Subgroups — C₂×#20295668 #28376479

Order 13952.

1900 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13182029	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295668	C₂	1	16	yes	maximal, direct product factor
4	C₂×#16812685	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16812684	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16812677	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16812674	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16812672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812671	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16812669	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16812669	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16812661	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#16812659	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#13182006	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂²⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#123×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
19	C₂×#13182029	—	2	8	no	—
20	C₂×#13182029	C₂²	1	8	yes	complemented
21	C₂²×Dic₂₁₈	C₂²	1	1	yes	characteristic, complemented
22	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
23	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
24	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
26	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
28	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic, complemented
29	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic, complemented
30	C₂×D₄×D₁₀₉	—	2	8	no	—
31	C₂×D₄×D₁₀₉	C₂²	1	8	yes	complemented
32	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
33	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
34	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
35	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
36	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
37	#16812685	C₂²	1	8	yes	complemented
38	#16812684	C₂²	1	8	yes	complemented
39	#16812677	C₂²	1	8	yes	complemented
40	#16812674	C₂²	1	8	yes	complemented
41	#16812672	C₂²	1	8	yes	complemented
42	#16812671	C₂²	1	8	yes	complemented
43	#16812669	C₂²	1	8	yes	complemented
44	#16812669	C₂²	1	8	yes	complemented
45	#16812661	C₂²	1	8	yes	complemented
46	#16812659	C₂²	1	8	yes	complemented
47	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
48	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
50	C₂×#13182006	C₂²	1	1	yes	characteristic
51	C₂×#13182006	C₂²	1	1	yes	characteristic
52	C₂×#13182006	C₂²	1	1	yes	characteristic
53	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
54	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
55	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
56	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
57	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
58	C₂×#13182006	C₂²	1	8	yes	complemented
59	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
60	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
61	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
62	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
63	#123×C₁₀₉	C₂²	1	8	yes	complemented
64	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
65	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
67	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
69	C₂³×D₁₀₉	—	2	4	no	—
70	C₂³×D₁₀₉	—	2	4	no	—
71	C₂³×D₁₀₉	—	2	4	no	—
72	C₂³×D₁₀₉	—	2	4	no	—
73	C₂³×D₁₀₉	—	4	2	no	—
74	C₂³×D₁₀₉	—	4	2	no	—
75	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
76	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
77	C₂³×D₁₀₉	C₂³	1	4	yes	—
78	C₂³×D₁₀₉	C₂³	1	4	yes	—
79	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
80	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
81	#13182029	—	2	32	no	—
82	C₂×Dic₂₁₈	—	2	4	no	—
83	C₂×Dic₂₁₈	C₂³	1	4	yes	complemented
84	C₂×D₄₃₆	—	2	4	no	—
85	C₂×D₄₃₆	—	2	4	no	—
86	C₂×D₄₃₆	—	2	4	no	—
87	C₂×D₄₃₆	—	4	2	no	—
88	C₂×D₄₃₆	C₂³	1	4	yes	—
89	C₂×D₄₃₆	C₂³	1	4	yes	—
90	C₂×D₄₃₆	C₂³	1	4	yes	complemented
91	C₂×D₄₃₆	C₂³	1	4	yes	complemented
92	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
93	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
94	C₂×C₂²⋊D₁₀₉	—	4	2	no	—
95	C₂×C₂²⋊D₁₀₉	D₄	1	8	yes	complemented
96	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
97	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
98	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	complemented
99	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	complemented
100	D₄×D₁₀₉	—	2	32	no	—
101	C₂²×Dic₁₀₉	—	4	2	no	—
102	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
103	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
105	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²×Dic₁₀₉	C₂³	1	4	yes	complemented
107	C₂×C₄×D₁₀₉	—	2	4	no	—
108	C₂×C₄×D₁₀₉	—	2	4	no	—
109	C₂×C₄×D₁₀₉	—	2	4	no	—
110	C₂×C₄×D₁₀₉	—	4	2	no	—
111	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
112	C₂×C₄×D₁₀₉	C₂³	1	4	yes	complemented
113	C₂×C₄×D₁₀₉	C₂³	1	4	yes	complemented
114	C₂×C₄×D₁₀₉	C₂³	1	4	yes	complemented
115	#13182008	C₂³	1	4	yes	complemented
116	#13182008	C₂³	1	4	yes	complemented
117	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	complemented
118	#13182006	C₂³	1	4	yes	—
119	#13182006	C₂³	1	4	yes	—
120	#13182006	C₂³	1	4	yes	—
121	#13182006	C₂³	1	4	yes	complemented
122	#13182006	C₂³	1	4	yes	complemented
123	#13182006	C₂³	1	4	yes	complemented
124	#13182006	C₂³	1	4	yes	complemented
125	#13182006	C₂³	1	4	yes	complemented
126	C₄×Dic₁₀₉	C₂³	1	4	yes	complemented
127	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
128	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
129	C₂×D₄×C₁₀₉	—	2	4	no	—
130	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
131	C₂²×C₄×C₁₀₉	—	2	4	no	—
132	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
133	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
134	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
135	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
136	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
137	C₂²×C₄×C₁₀₉	C₂³	1	4	yes	complemented
138	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
139	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
140	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
141	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
142	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
143	C₂³×C₁₀₉	—	2	2	no	—
144	C₂³×C₁₀₉	—	2	2	no	—
145	C₂³×C₁₀₉	—	2	2	no	—
146	C₂³×C₁₀₉	—	2	2	no	—
147	C₂³×C₁₀₉	—	2	2	no	—
148	C₂³×C₁₀₉	C₂×D₄	1	4	yes	—
149	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
150	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
151	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
152	C₂²×D₁₀₉	—	2	2	no	—
153	C₂²×D₁₀₉	—	2	2	no	—
154	C₂²×D₁₀₉	—	2	2	no	—
155	C₂²×D₁₀₉	—	2	2	no	—
156	C₂²×D₁₀₉	—	2	2	no	—
157	C₂²×D₁₀₉	—	2	8	no	—
158	C₂²×D₁₀₉	—	2	2	no	—
159	C₂²×D₁₀₉	—	2	2	no	—
160	C₂²×D₁₀₉	—	2	2	no	—
161	C₂²×D₁₀₉	—	2	2	no	—
162	C₂²×D₁₀₉	—	2	2	no	—
163	C₂²×D₁₀₉	—	2	8	no	—
164	C₂²×D₁₀₉	—	2	8	no	—
165	C₂²×D₁₀₉	—	2	8	no	—
166	C₂²×D₁₀₉	—	4	4	no	—
167	C₂²×D₁₀₉	—	4	4	no	—
168	C₂²×D₁₀₉	—	4	16	no	—
169	C₂²×D₁₀₉	—	4	16	no	—
170	C₂²×D₁₀₉	C₂×D₄	1	4	yes	—
171	C₂²×D₁₀₉	#41	1	4	yes	complemented
172	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
173	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
174	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
175	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
176	D₄×C₁₀₉	—	2	8	no	—
177	C₂×C₄×C₁₀₉	—	2	2	no	—
178	C₂×C₄×C₁₀₉	—	2	2	no	—
179	C₂×C₄×C₁₀₉	—	2	8	no	—
180	C₂×C₄×C₁₀₉	—	2	2	no	—
181	C₂×C₄×C₁₀₉	—	2	2	no	—
182	C₂×C₄×C₁₀₉	—	2	2	no	—
183	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
184	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
185	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
186	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
187	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
188	C₂²⋊D₁₀₉	—	2	8	no	—
189	C₂²⋊D₁₀₉	—	2	8	no	—
190	C₂²⋊D₁₀₉	—	4	4	no	—
191	C₂²⋊D₁₀₉	C₂×D₄	1	16	yes	complemented
192	C₂×Dic₁₀₉	—	2	2	no	—
193	C₂×Dic₁₀₉	—	2	2	no	—
194	C₂×Dic₁₀₉	—	2	2	no	—
195	C₂×Dic₁₀₉	—	4	4	no	—
196	C₂×Dic₁₀₉	C₂×D₄	1	4	yes	—
197	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
198	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
199	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
200	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
201	D₄₃₆	—	2	8	no	—
202	D₄₃₆	—	2	8	no	—
203	D₄₃₆	—	2	8	no	—
204	D₄₃₆	—	4	4	no	—
205	C₄×D₁₀₉	—	2	8	no	—
206	C₄×D₁₀₉	—	2	8	no	—
207	C₄×D₁₀₉	—	2	8	no	—
208	C₄×D₁₀₉	—	4	4	no	—
209	Dic₂₁₈	—	2	8	no	—
210	C₂²×C₁₀₉	—	2	8	no	—
211	C₂²×C₁₀₉	—	2	2	no	—
212	C₂²×C₁₀₉	—	2	2	no	—
213	C₂²×C₁₀₉	—	2	2	no	—
214	C₂²×C₁₀₉	—	2	2	no	—
215	C₂²×C₁₀₉	—	2	2	no	—
216	C₂²×C₁₀₉	—	4	4	no	—
217	C₂²×C₁₀₉	C₂²×D₄	1	4	yes	—
218	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
219	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
220	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
221	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
222	C₂×D₁₀₉	—	2	8	no	—
223	C₂×D₁₀₉	—	2	8	no	—
224	C₂×D₁₀₉	—	2	2	no	—
225	C₂×D₁₀₉	—	2	2	no	—
226	C₂×D₁₀₉	—	2	2	no	—
227	C₂×D₁₀₉	—	2	2	no	—
228	C₂×D₁₀₉	—	2	2	no	—
229	C₂×D₁₀₉	—	2	2	no	—
230	C₂×D₁₀₉	—	2	2	no	—
231	C₂×D₁₀₉	—	2	2	no	—
232	C₂×D₁₀₉	—	2	2	no	—
233	C₂×D₁₀₉	—	2	2	no	—
234	C₂×D₁₀₉	—	4	16	no	—
235	C₂×D₁₀₉	—	4	16	no	—
236	C₂×D₁₀₉	—	4	4	no	—
237	C₂×D₁₀₉	—	4	4	no	—
238	C₂×D₁₀₉	C₂²×D₄	1	4	yes	—
239	C₂×D₁₀₉	C₂×#41	1	4	yes	complemented
240	C₄×C₁₀₉	—	2	2	no	—
241	C₄×C₁₀₉	—	2	2	no	—
242	C₄×C₁₀₉	—	2	2	no	—
243	C₄×C₁₀₉	—	2	2	no	—
244	C₄×C₁₀₉	—	2	2	no	—
245	Dic₁₀₉	—	2	2	no	—
246	Dic₁₀₉	—	2	2	no	—
247	Dic₁₀₉	—	2	2	no	—
248	Dic₁₀₉	C₂²×D₄	1	4	yes	—
249	C₂×C₁₀₉	—	2	4	no	—
250	C₂×C₁₀₉	—	4	2	no	—
251	C₂×C₁₀₉	#546	1	4	yes	complemented
252	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
253	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
254	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
255	D₁₀₉	—	2	4	no	—
256	D₁₀₉	—	2	4	no	—
257	D₁₀₉	—	4	2	no	—
258	D₁₀₉	—	4	2	no	—
259	C₂×#546	—	109	1	no	maximal, Sylow
260	C₁₀₉	C₂×#546	1	1	yes	characteristic, complemented, Sylow
261	C₂²×#41	—	109	1	no	—
262	C₂³×D₄	—	109	1	no	—
263	#546	—	109	16	no	—
264	C₂×#124	—	109	1	no	—
265	C₂×#123	—	109	1	no	—
266	C₂×#123	D₁₀₉	1	1	yes	characteristic, complemented
267	C₂×C₂²⋊Q₈	—	109	1	no	—
268	C₂×#121	—	109	1	no	—
269	C₂×#121	—	109	1	no	—
270	C₂×#121	—	109	1	no	—
271	C₂×#120	—	109	1	no	—
272	C₂×#120	—	109	1	no	—
273	C₂×C₄×D₄	—	109	1	no	—
274	C₂×C₄×D₄	—	109	1	no	—
275	C₂²×C₂²⋊C₄	—	109	1	no	—
276	C₂²×C₂²⋊C₄	—	109	1	no	—
277	C₂⁵	—	109	1	no	—
278	C₂⁵	—	109	1	no	—
279	C₂×#41	—	109	8	no	—
280	C₂×#41	—	218	8	no	—
281	C₂²×Q₈	—	109	1	no	—
282	C₂²×D₄	—	109	1	no	—
283	C₂²×D₄	—	109	1	no	—
284	C₂²×D₄	—	109	1	no	—
285	C₂²×D₄	—	109	1	no	—
286	C₂²×D₄	—	109	8	no	—
287	C₂²×D₄	—	109	1	no	—
288	C₂²×D₄	—	109	1	no	—
289	C₂²×D₄	—	109	1	no	—
290	C₂²×D₄	—	109	1	no	—
291	C₂²×D₄	—	218	8	no	—
292	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
293	C₂³×C₄	—	109	1	no	—
294	C₂³×C₄	—	109	1	no	—
295	C₂³×C₄	—	109	1	no	—
296	C₂³×C₄	—	109	1	no	—
297	C₂³×C₄	—	109	1	no	—
298	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
299	#124	—	109	8	no	—
300	#123	—	109	8	no	—
301	#123	C₂×D₁₀₉	1	8	yes	complemented
302	C₂²⋊Q₈	—	109	8	no	—
303	#121	—	109	8	no	—
304	#121	—	109	8	no	—
305	#121	—	109	8	no	—
306	#120	—	109	8	no	—
307	#120	—	109	8	no	—
308	C₄×D₄	—	109	8	no	—
309	C₄×D₄	—	109	8	no	—
310	C₂×C₄⋊C₄	—	109	1	no	—
311	C₂×C₄⋊C₄	—	109	1	no	—
312	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
313	C₂×C₄⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
314	C₂×C₂²⋊C₄	—	109	1	no	—
315	C₂×C₂²⋊C₄	—	109	1	no	—
316	C₂×C₂²⋊C₄	—	109	1	no	—
317	C₂×C₂²⋊C₄	—	109	1	no	—
318	C₂×C₂²⋊C₄	—	109	1	no	—
319	C₂×C₂²⋊C₄	—	109	1	no	—
320	C₂×C₂²⋊C₄	—	109	1	no	—
321	C₂×C₂²⋊C₄	—	109	1	no	—
322	C₂×C₂²⋊C₄	—	109	1	no	—
323	C₂×C₂²⋊C₄	—	109	8	no	—
324	C₂×C₂²⋊C₄	—	109	8	no	—
325	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
326	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
327	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
328	C₂×C₄²	—	109	1	no	—
329	C₂⁴	—	109	1	no	—
330	C₂⁴	—	109	1	no	—
331	C₂⁴	—	109	4	no	—
332	C₂⁴	—	109	4	no	—
333	C₂⁴	—	109	1	no	—
334	C₂⁴	—	109	1	no	—
335	C₂⁴	—	436	2	no	—
336	C₂⁴	—	436	2	no	—
337	C₂⁴	—	218	4	no	—
338	C₂⁴	—	218	4	no	—
339	C₂⁴	—	218	4	no	—
340	C₂⁴	—	218	4	no	—
341	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
342	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
343	#41	—	218	32	no	—
344	C₂×Q₈	—	109	4	no	—
345	C₂×Q₈	—	218	4	no	—
346	C₂×D₄	—	2	4	no	—
347	C₂×D₄	—	109	8	no	—
348	C₂×D₄	—	109	4	no	—
349	C₂×D₄	—	109	4	no	—
350	C₂×D₄	—	109	4	no	—
351	C₂×D₄	—	109	4	no	—
352	C₂×D₄	—	109	4	no	—
353	C₂×D₄	—	109	4	no	—
354	C₂×D₄	—	109	4	no	—
355	C₂×D₄	—	109	4	no	—
356	C₂×D₄	—	436	2	no	—
357	C₂×D₄	—	436	2	no	—
358	C₂×D₄	—	218	4	no	—
359	C₂×D₄	—	218	4	no	—
360	C₂×D₄	—	218	4	no	—
361	C₂×D₄	—	218	4	no	—
362	C₂×D₄	—	218	32	no	—
363	C₂×D₄	—	218	4	no	—
364	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
365	C₂²×C₄	—	2	4	no	—
366	C₂²×C₄	—	109	1	no	—
367	C₂²×C₄	—	109	4	no	—
368	C₂²×C₄	—	109	1	no	—
369	C₂²×C₄	—	109	1	no	—
370	C₂²×C₄	—	109	1	no	—
371	C₂²×C₄	—	109	4	no	—
372	C₂²×C₄	—	109	4	no	—
373	C₂²×C₄	—	109	4	no	—
374	C₂²×C₄	—	109	4	no	—
375	C₂²×C₄	—	436	2	no	—
376	C₂²×C₄	—	436	2	no	—
377	C₂²×C₄	—	218	4	no	—
378	C₂²×C₄	—	218	4	no	—
379	C₂²×C₄	—	218	4	no	—
380	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
381	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
382	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
383	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
384	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
385	C₂²×C₄	C₂²×D₁₀₉	1	4	yes	complemented
386	C₄⋊C₄	—	109	4	no	—
387	C₄⋊C₄	—	109	4	no	—
388	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
389	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
390	C₂²⋊C₄	—	109	4	no	—
391	C₂²⋊C₄	—	109	4	no	—
392	C₂²⋊C₄	—	109	4	no	—
393	C₂²⋊C₄	—	109	4	no	—
394	C₂²⋊C₄	—	109	4	no	—
395	C₂²⋊C₄	—	109	4	no	—
396	C₂²⋊C₄	—	109	4	no	—
397	C₂²⋊C₄	—	109	4	no	—
398	C₂²⋊C₄	—	109	4	no	—
399	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
400	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
401	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
402	C₄²	—	109	4	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	2	2	no	—
406	C₂³	—	2	2	no	—
407	C₂³	—	2	2	no	—
408	C₂³	—	109	4	no	—
409	C₂³	—	109	4	no	—
410	C₂³	—	109	2	no	—
411	C₂³	—	109	2	no	—
412	C₂³	—	109	2	no	—
413	C₂³	—	109	2	no	—
414	C₂³	—	436	16	no	—
415	C₂³	—	436	4	no	—
416	C₂³	—	436	4	no	—
417	C₂³	—	436	16	no	—
418	C₂³	—	218	8	no	—
419	C₂³	—	218	8	no	—
420	C₂³	—	218	8	no	—
421	C₂³	—	218	2	no	—
422	C₂³	—	218	8	no	—
423	C₂³	—	218	2	no	—
424	C₂³	—	218	2	no	—
425	C₂³	—	218	2	no	—
426	C₂³	—	218	2	no	—
427	C₂³	—	218	2	no	—
428	C₂³	—	218	2	no	—
429	C₂³	—	218	2	no	—
430	C₂³	—	218	2	no	—
431	C₂³	—	218	2	no	—
432	C₂³	D₄×D₁₀₉	1	4	yes	—
433	C₂³	C₂³×D₁₀₉	1	2	yes	—
434	C₂³	C₂³×D₁₀₉	1	2	yes	—
435	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
436	Q₈	—	218	8	no	—
437	D₄	—	2	8	no	—
438	D₄	—	109	16	no	—
439	D₄	—	436	4	no	—
440	D₄	—	436	4	no	—
441	D₄	—	218	8	no	—
442	D₄	—	218	8	no	—
443	D₄	—	218	8	no	—
444	D₄	—	218	8	no	—
445	D₄	—	218	8	no	—
446	C₂×C₄	—	2	2	no	—
447	C₂×C₄	—	2	2	no	—
448	C₂×C₄	—	2	8	no	—
449	C₂×C₄	—	2	2	no	—
450	C₂×C₄	—	2	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	109	4	no	—
453	C₂×C₄	—	109	2	no	—
454	C₂×C₄	—	109	2	no	—
455	C₂×C₄	—	109	2	no	—
456	C₂×C₄	—	109	2	no	—
457	C₂×C₄	—	436	4	no	—
458	C₂×C₄	—	436	4	no	—
459	C₂×C₄	—	218	8	no	—
460	C₂×C₄	—	218	8	no	—
461	C₂×C₄	—	218	8	no	—
462	C₂×C₄	—	218	2	no	—
463	C₂×C₄	—	218	2	no	—
464	C₂×C₄	—	218	2	no	—
465	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
466	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
467	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
468	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
469	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
470	C₂²	—	2	2	no	—
471	C₂²	—	2	2	no	—
472	C₂²	—	2	8	no	—
473	C₂²	—	2	2	no	—
474	C₂²	—	2	2	no	—
475	C₂²	—	2	2	no	—
476	C₂²	—	4	4	no	—
477	C₂²	—	109	4	no	—
478	C₂²	—	109	4	no	—
479	C₂²	—	436	16	no	—
480	C₂²	—	436	16	no	—
481	C₂²	—	436	4	no	—
482	C₂²	—	436	4	no	—
483	C₂²	—	218	2	no	—
484	C₂²	—	218	2	no	—
485	C₂²	—	218	2	no	—
486	C₂²	—	218	2	no	—
487	C₂²	—	218	2	no	—
488	C₂²	—	218	2	no	—
489	C₂²	—	218	2	no	—
490	C₂²	—	218	2	no	—
491	C₂²	—	218	2	no	—
492	C₂²	—	218	2	no	—
493	C₂²	—	218	8	no	—
494	C₂²	—	218	8	no	—
495	C₂²	C₂×D₄×D₁₀₉	1	4	yes	—
496	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
497	C₂²	#41×D₁₀₉	1	2	yes	—
498	C₂²	#16812745	1	2	yes	—
499	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
500	C₄	—	2	2	no	—
501	C₄	—	2	2	no	—
502	C₄	—	2	2	no	—
503	C₄	—	2	2	no	—
504	C₄	—	2	2	no	—
505	C₄	—	109	4	no	—
506	C₄	—	218	2	no	—
507	C₄	—	218	2	no	—
508	C₄	—	218	2	no	—
509	C₂	—	2	4	no	—
510	C₂	—	4	2	no	—
511	C₂	—	436	2	no	—
512	C₂	—	436	2	no	—
513	C₂	—	218	4	no	—
514	C₂	—	218	4	no	—
515	C₂	#20295668	1	4	yes	direct product factor
516	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
517	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
518	C₂	C₂×#16812745	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂³×D₁₀₉
commutator, cosocle	C₂²×C₁₀₉	C₂⁵
Frattini	C₂²	C₂⁴×D₁₀₉
Fitting	C₂×#123×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂×#123	D₁₀₉
hypocenter	C₁₀₉	C₂×#546
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×#123

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂²×#13182029	C₂	semidirect
2. C₂²×D₄×D₁₀₉	C₂	semidirect
3. #20295668	C₂	direct (N × H)
4. C₂×#16812685	C₂	semidirect
5. C₂×#16812684	C₂	semidirect
6. C₂×#16812677	C₂	semidirect
7. C₂×#16812674	C₂	semidirect
8. C₂×#16812672	C₂	semidirect
9. C₂×#16812671	C₂	semidirect
10. C₂×#16812669	C₂	semidirect
11. C₂×#16812669	C₂	semidirect
12. C₂×#16812661	C₂	semidirect
13. C₂×#16812659	C₂	semidirect
14. C₂²×#13182006	C₂	semidirect
15. C₂×C₂²⋊C₄×D₁₀₉	C₂	semidirect
16. C₂×#123×C₁₀₉	C₂	semidirect
20. C₂×#13182029	C₂²	semidirect
21. C₂²×Dic₂₁₈	C₂²	semidirect
24. C₂²×D₄₃₆	C₂²	semidirect
25. C₂²×D₄₃₆	C₂²	semidirect
28. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
29. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
31. C₂×D₄×D₁₀₉	C₂²	semidirect
32. C₂³×Dic₁₀₉	C₂²	semidirect
34. C₂²×C₄×D₁₀₉	C₂²	semidirect
35. C₂²×C₄×D₁₀₉	C₂²	semidirect
36. C₂²×C₄×D₁₀₉	C₂²	semidirect
37. #16812685	C₂²	semidirect
38. #16812684	C₂²	semidirect
39. #16812677	C₂²	semidirect
40. #16812674	C₂²	semidirect
41. #16812672	C₂²	semidirect
42. #16812671	C₂²	semidirect
43. #16812669	C₂²	semidirect
44. #16812669	C₂²	semidirect
45. #16812661	C₂²	semidirect
46. #16812659	C₂²	semidirect
47. C₂×#13182008	C₂²	semidirect
48. C₂×#13182008	C₂²	semidirect
49. C₂×C₂²⋊Dic₁₀₉	C₂²	semidirect
53. C₂×#13182006	C₂²	semidirect
54. C₂×#13182006	C₂²	semidirect
55. C₂×#13182006	C₂²	semidirect
56. C₂×#13182006	C₂²	semidirect
57. C₂×#13182006	C₂²	semidirect
58. C₂×#13182006	C₂²	semidirect
59. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
60. C₂×C₄×Dic₁₀₉	C₂²	semidirect
62. C₂³×C₄×C₁₀₉	C₂²	semidirect
63. #123×C₁₀₉	C₂²	semidirect
64. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
65. C₂×C₄⋊C₄×C₁₀₉	C₂²	semidirect
66. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
67. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
68. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
83. C₂×Dic₂₁₈	C₂³	semidirect
90. C₂×D₄₃₆	C₂³	semidirect
91. C₂×D₄₃₆	C₂³	semidirect
95. C₂×C₂²⋊D₁₀₉	D₄	semidirect
98. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
99. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
106. C₂²×Dic₁₀₉	C₂³	semidirect
112. C₂×C₄×D₁₀₉	C₂³	semidirect
113. C₂×C₄×D₁₀₉	C₂³	semidirect
114. C₂×C₄×D₁₀₉	C₂³	semidirect
115. #13182008	C₂³	semidirect
116. #13182008	C₂³	semidirect
117. C₂²⋊Dic₁₀₉	C₂³	semidirect
121. #13182006	C₂³	semidirect
122. #13182006	C₂³	semidirect
123. #13182006	C₂³	semidirect
124. #13182006	C₂³	semidirect
125. #13182006	C₂³	semidirect
126. C₄×Dic₁₀₉	C₂³	semidirect
137. C₂²×C₄×C₁₀₉	C₂³	semidirect
138. C₄⋊C₄×C₁₀₉	C₂³	semidirect
139. C₄⋊C₄×C₁₀₉	C₂³	semidirect
140. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
141. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
142. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
171. C₂²×D₁₀₉	#41	semidirect
191. C₂²⋊D₁₀₉	C₂×D₄	semidirect
239. C₂×D₁₀₉	C₂×#41	semidirect
251. C₂×C₁₀₉	#546	semidirect
260. C₁₀₉	C₂×#546	semidirect
266. C₂×#123	D₁₀₉	semidirect
298. C₂³×C₄	C₂×D₁₀₉	semidirect
301. #123	C₂×D₁₀₉	semidirect
312. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
313. C₂×C₄⋊C₄	C₂×D₁₀₉	semidirect
325. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
326. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
327. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
385. C₂²×C₄	C₂²×D₁₀₉	semidirect
388. C₄⋊C₄	C₂²×D₁₀₉	semidirect
389. C₄⋊C₄	C₂²×D₁₀₉	semidirect
399. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
400. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
401. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
515. C₂	#20295668	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#13182029	2
C₂²×D₄×D₁₀₉	2
#20295668	2
C₂×#16812685	2
C₂×#16812684	2
C₂×#16812677	2
C₂×#16812674	2
C₂×#16812672	2
C₂×#16812671	2
C₂×#16812669	2
C₂×#16812661	2
C₂×#16812659	2
C₂²×#13182006	2
C₂×C₂²⋊C₄×D₁₀₉	2
C₂×#123×C₁₀₉	2
C₂×#546	109

Subgroups — C2×#20295668 #28376479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295668 #28376479