Subgroups — C₂×#20295610 #28376369

Order 13952.

1336 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295610	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16812693	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812692	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812688	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16812683	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16812681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#16812675	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16812672	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812672	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16812671	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16812669	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16812667	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₄×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×#16812659	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₂²⋊C₄×D₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#124×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
17	C₂⁴×D₁₀₉	C₂²	1	1	yes	characteristic
18	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
19	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
20	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
21	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
22	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₀₉	C₂²	1	1	yes	characteristic, complemented
24	C₂³×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
26	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₁₀₉	C₂²	1	1	yes	characteristic, complemented
28	#16812693	C₂²	1	8	yes	complemented
29	#16812692	C₂²	1	8	yes	complemented
30	#16812688	C₂²	1	8	yes	complemented
31	#16812683	C₂²	1	8	yes	complemented
32	#16812681	C₂²	1	8	yes	complemented
33	#16812675	C₂²	1	8	yes	complemented
34	#16812672	C₂²	1	8	yes	complemented
35	#16812672	C₂²	1	8	yes	complemented
36	#16812671	C₂²	1	8	yes	complemented
37	#16812669	C₂²	1	8	yes	complemented
38	#16812667	C₂²	1	8	yes	complemented
39	C₄×D₄₃₆	C₂²	1	8	yes	complemented
40	#16812659	C₂²	1	8	yes	complemented
41	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
42	C₂×C₄⋊Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
43	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
44	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
45	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
46	C₂×#13182008	C₂²	1	1	yes	characteristic, complemented
47	C₂×C₂²⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊Dic₁₀₉	C₂²	1	1	yes	characteristic
49	C₂×#13182006	C₂²	1	1	yes	characteristic
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
54	C₂×#13182006	C₂²	1	1	yes	characteristic
55	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
56	C₂×#13182006	C₂²	1	1	yes	characteristic, complemented
57	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
58	C₂×C₄×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
59	C₂²×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
60	C₂²×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
61	#124×C₁₀₉	C₂²	1	8	yes	complemented
62	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
63	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
65	C₂×C₂²⋊C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
67	C₂³×D₁₀₉	—	2	4	no	—
68	C₂³×D₁₀₉	—	4	2	no	—
69	C₂³×D₁₀₉	—	4	2	no	—
70	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
71	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
72	C₂³×D₁₀₉	C₂³	1	1	yes	characteristic
73	C₂³×D₁₀₉	C₂³	1	4	yes	—
74	C₂×D₄₃₆	—	2	4	no	—
75	C₂×D₄₃₆	—	4	2	no	—
76	C₂×D₄₃₆	C₂³	1	4	yes	—
77	C₂×D₄₃₆	C₂³	1	4	yes	—
78	C₂×C₂²⋊D₁₀₉	—	2	4	no	—
79	C₂×C₂²⋊D₁₀₉	—	4	2	no	—
80	C₂×C₂²⋊D₁₀₉	—	4	2	no	—
81	C₂×C₂²⋊D₁₀₉	—	4	2	no	—
82	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
83	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
84	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
85	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	complemented
86	C₂²×Dic₁₀₉	—	4	2	no	—
87	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
88	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
89	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
90	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
91	C₂²×Dic₁₀₉	C₂³	1	1	yes	characteristic
92	C₂²×Dic₁₀₉	C₂³	1	4	yes	complemented
93	C₂×C₄×D₁₀₉	—	4	2	no	—
94	C₂×C₄×D₁₀₉	—	4	2	no	—
95	C₂×C₄×D₁₀₉	—	4	2	no	—
96	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
97	C₂×C₄×D₁₀₉	C₂³	1	4	yes	—
98	C₂×C₄×D₁₀₉	C₂³	1	4	yes	complemented
99	C₄⋊Dic₁₀₉	C₂³	1	4	yes	complemented
100	C₄⋊Dic₁₀₉	C₂³	1	4	yes	complemented
101	#13182008	C₂³	1	4	yes	complemented
102	#13182008	C₂³	1	4	yes	complemented
103	#13182008	C₂³	1	4	yes	complemented
104	#13182008	C₂³	1	4	yes	complemented
105	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
106	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
107	#13182006	C₂³	1	4	yes	—
108	#13182006	C₂³	1	4	yes	—
109	#13182006	C₂³	1	4	yes	—
110	#13182006	C₂³	1	4	yes	—
111	#13182006	C₂³	1	4	yes	—
112	#13182006	C₂³	1	4	yes	—
113	#13182006	C₂³	1	4	yes	complemented
114	#13182006	C₂³	1	4	yes	complemented
115	C₄×Dic₁₀₉	C₂³	1	4	yes	complemented
116	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
117	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
118	C₂×Q₈×C₁₀₉	—	4	2	no	—
119	C₂×Q₈×C₁₀₉	C₂³	1	4	yes	complemented
120	C₂×D₄×C₁₀₉	—	4	2	no	—
121	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
122	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
123	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
124	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
125	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
126	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
127	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
129	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
130	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
131	C₄²×C₁₀₉	C₂³	1	4	yes	complemented
132	C₂³×C₁₀₉	—	2	2	no	—
133	C₂³×C₁₀₉	—	2	2	no	—
134	C₂³×C₁₀₉	—	2	2	no	—
135	C₂³×C₁₀₉	—	2	2	no	—
136	C₂³×C₁₀₉	—	2	2	no	—
137	C₂³×C₁₀₉	—	2	2	no	—
138	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
139	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
140	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
141	C₂²×D₁₀₉	—	2	2	no	—
142	C₂²×D₁₀₉	—	2	2	no	—
143	C₂²×D₁₀₉	—	2	2	no	—
144	C₂²×D₁₀₉	—	2	2	no	—
145	C₂²×D₁₀₉	—	2	2	no	—
146	C₂²×D₁₀₉	—	2	2	no	—
147	C₂²×D₁₀₉	—	2	2	no	—
148	C₂²×D₁₀₉	—	2	2	no	—
149	C₂²×D₁₀₉	—	2	8	no	—
150	C₂²×D₁₀₉	—	4	4	no	—
151	C₂²×D₁₀₉	—	4	4	no	—
152	C₂²×D₁₀₉	—	4	16	no	—
153	C₂²×D₁₀₉	#41	1	4	yes	—
154	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
155	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
156	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
157	Q₈×C₁₀₉	—	4	4	no	—
158	D₄×C₁₀₉	—	4	4	no	—
159	C₂×C₄×C₁₀₉	—	2	2	no	—
160	C₂×C₄×C₁₀₉	—	2	2	no	—
161	C₂×C₄×C₁₀₉	—	2	2	no	—
162	C₂×C₄×C₁₀₉	—	2	2	no	—
163	C₂×C₄×C₁₀₉	—	2	2	no	—
164	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
165	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
166	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
167	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
168	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
169	C₂²⋊D₁₀₉	—	2	8	no	—
170	C₂²⋊D₁₀₉	—	4	4	no	—
171	C₂²⋊D₁₀₉	—	4	4	no	—
172	C₂²⋊D₁₀₉	—	4	4	no	—
173	C₂×Dic₁₀₉	—	2	2	no	—
174	C₂×Dic₁₀₉	—	2	2	no	—
175	C₂×Dic₁₀₉	—	2	2	no	—
176	C₂×Dic₁₀₉	—	2	2	no	—
177	C₂×Dic₁₀₉	—	2	2	no	—
178	C₂×Dic₁₀₉	—	4	4	no	—
179	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
180	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
181	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
182	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
183	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
184	D₄₃₆	—	2	8	no	—
185	D₄₃₆	—	4	4	no	—
186	C₄×D₁₀₉	—	4	4	no	—
187	C₄×D₁₀₉	—	4	4	no	—
188	C₄×D₁₀₉	—	4	4	no	—
189	C₂²×C₁₀₉	—	2	2	no	—
190	C₂²×C₁₀₉	—	2	2	no	—
191	C₂²×C₁₀₉	—	2	2	no	—
192	C₂²×C₁₀₉	—	2	2	no	—
193	C₂²×C₁₀₉	—	2	2	no	—
194	C₂²×C₁₀₉	—	2	2	no	—
195	C₂²×C₁₀₉	—	4	4	no	—
196	C₂²×C₁₀₉	—	4	4	no	—
197	C₂²×C₁₀₉	C₂×#41	1	2	yes	—
198	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
199	C₂²×C₁₀₉	H(5,𝔽₂)	1	2	yes	—
200	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
201	C₂×D₁₀₉	—	2	8	no	—
202	C₂×D₁₀₉	—	2	2	no	—
203	C₂×D₁₀₉	—	2	2	no	—
204	C₂×D₁₀₉	—	2	2	no	—
205	C₂×D₁₀₉	—	2	2	no	—
206	C₂×D₁₀₉	—	2	2	no	—
207	C₂×D₁₀₉	—	2	2	no	—
208	C₂×D₁₀₉	—	2	2	no	—
209	C₂×D₁₀₉	—	2	2	no	—
210	C₂×D₁₀₉	—	4	16	no	—
211	C₂×D₁₀₉	—	4	4	no	—
212	C₂×D₁₀₉	—	4	4	no	—
213	C₂×D₁₀₉	C₂×#41	1	4	yes	—
214	C₄×C₁₀₉	—	2	2	no	—
215	C₄×C₁₀₉	—	2	2	no	—
216	C₄×C₁₀₉	—	2	2	no	—
217	C₄×C₁₀₉	—	2	2	no	—
218	C₄×C₁₀₉	—	2	2	no	—
219	Dic₁₀₉	—	2	2	no	—
220	Dic₁₀₉	—	2	2	no	—
221	Dic₁₀₉	—	2	2	no	—
222	Dic₁₀₉	—	2	2	no	—
223	Dic₁₀₉	—	2	2	no	—
224	C₂×C₁₀₉	—	4	2	no	—
225	C₂×C₁₀₉	—	4	2	no	—
226	C₂×C₁₀₉	#538	1	4	yes	complemented
227	C₂×C₁₀₉	C₂²×#41	1	1	yes	characteristic
228	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
229	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
230	D₁₀₉	—	2	4	no	—
231	D₁₀₉	—	4	2	no	—
232	D₁₀₉	—	4	2	no	—
233	C₂×#538	—	109	1	no	maximal, Sylow
234	C₁₀₉	C₂×#538	1	1	yes	characteristic, complemented, Sylow
235	#538	—	109	16	no	—
236	C₂×#126	—	109	1	no	—
237	C₂×#126	—	109	1	no	—
238	C₂×#124	—	109	1	no	—
239	C₂×#124	D₁₀₉	1	1	yes	characteristic, complemented
240	C₂×#123	—	109	1	no	—
241	C₂×#123	—	109	1	no	—
242	C₂×C₂²⋊Q₈	—	109	1	no	—
243	C₂×#121	—	109	1	no	—
244	C₂×#121	—	109	1	no	—
245	C₂×#121	—	109	1	no	—
246	C₂×#120	—	109	1	no	—
247	C₂×#120	—	109	1	no	—
248	C₂×C₄×D₄	—	109	1	no	—
249	C₂×C₄×D₄	—	109	1	no	—
250	C₂²×C₂²⋊C₄	—	109	1	no	—
251	C₂⁵	—	109	1	no	—
252	C₂²×Q₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
253	C₂²×D₄	—	109	1	no	—
254	C₂²×D₄	—	109	1	no	—
255	C₂²×D₄	—	109	1	no	—
256	C₂²×D₄	—	109	1	no	—
257	C₂²×D₄	—	109	1	no	—
258	C₂²×D₄	—	109	1	no	—
259	C₂²×D₄	C₂×D₁₀₉	1	1	yes	characteristic
260	C₂³×C₄	—	109	1	no	—
261	C₂³×C₄	—	109	1	no	—
262	C₂³×C₄	—	109	1	no	—
263	C₂³×C₄	—	109	1	no	—
264	#126	—	109	8	no	—
265	#126	—	109	8	no	—
266	#124	—	109	8	no	—
267	#124	C₂×D₁₀₉	1	8	yes	complemented
268	#123	—	109	8	no	—
269	#123	—	109	8	no	—
270	C₂²⋊Q₈	—	109	8	no	—
271	#121	—	109	8	no	—
272	#121	—	109	8	no	—
273	#121	—	109	8	no	—
274	#120	—	109	8	no	—
275	#120	—	109	8	no	—
276	C₄×D₄	—	109	8	no	—
277	C₄×D₄	—	109	8	no	—
278	C₂×C₄⋊C₄	—	109	1	no	—
279	C₂×C₄⋊C₄	—	109	1	no	—
280	C₂×C₄⋊C₄	—	109	1	no	—
281	C₂×C₄⋊C₄	—	109	1	no	—
282	C₂×C₄⋊C₄	—	109	1	no	—
283	C₂×C₄⋊C₄	—	109	1	no	—
284	C₂×C₂²⋊C₄	—	109	1	no	—
285	C₂×C₂²⋊C₄	—	109	1	no	—
286	C₂×C₂²⋊C₄	—	109	1	no	—
287	C₂×C₂²⋊C₄	—	109	1	no	—
288	C₂×C₂²⋊C₄	—	109	1	no	—
289	C₂×C₂²⋊C₄	—	109	1	no	—
290	C₂×C₂²⋊C₄	—	109	1	no	—
291	C₂×C₂²⋊C₄	—	109	1	no	—
292	C₂×C₂²⋊C₄	—	109	1	no	—
293	C₂×C₂²⋊C₄	—	109	1	no	—
294	C₂×C₂²⋊C₄	—	109	8	no	—
295	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
296	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic
297	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
298	C₂×C₂²⋊C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
299	C₂×C₄²	—	109	1	no	—
300	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic, complemented
301	C₂⁴	—	109	1	no	—
302	C₂⁴	—	109	4	no	—
303	C₂⁴	—	109	1	no	—
304	C₂⁴	—	109	1	no	—
305	C₂⁴	—	436	2	no	—
306	C₂⁴	—	436	2	no	—
307	C₂⁴	—	218	4	no	—
308	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
309	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
310	C₂×Q₈	—	4	2	no	—
311	C₂×Q₈	C₂²×D₁₀₉	1	4	yes	complemented
312	C₂×D₄	—	4	2	no	—
313	C₂×D₄	—	109	4	no	—
314	C₂×D₄	—	109	4	no	—
315	C₂×D₄	—	109	4	no	—
316	C₂×D₄	—	109	4	no	—
317	C₂×D₄	—	109	4	no	—
318	C₂×D₄	—	109	4	no	—
319	C₂×D₄	—	436	2	no	—
320	C₂×D₄	—	436	2	no	—
321	C₂×D₄	—	436	2	no	—
322	C₂×D₄	—	436	2	no	—
323	C₂×D₄	—	218	4	no	—
324	C₂×D₄	—	218	4	no	—
325	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
326	C₂²×C₄	—	109	4	no	—
327	C₂²×C₄	—	109	1	no	—
328	C₂²×C₄	—	109	1	no	—
329	C₂²×C₄	—	109	1	no	—
330	C₂²×C₄	—	109	1	no	—
331	C₂²×C₄	—	109	1	no	—
332	C₂²×C₄	—	109	4	no	—
333	C₂²×C₄	—	109	4	no	—
334	C₂²×C₄	—	109	4	no	—
335	C₂²×C₄	—	436	2	no	—
336	C₂²×C₄	—	436	2	no	—
337	C₂²×C₄	—	436	2	no	—
338	C₂²×C₄	—	436	2	no	—
339	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
340	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
341	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
342	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
343	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
344	C₄⋊C₄	—	109	4	no	—
345	C₄⋊C₄	—	109	4	no	—
346	C₄⋊C₄	—	109	4	no	—
347	C₄⋊C₄	—	109	4	no	—
348	C₄⋊C₄	—	109	4	no	—
349	C₄⋊C₄	—	109	4	no	—
350	C₂²⋊C₄	—	109	4	no	—
351	C₂²⋊C₄	—	109	4	no	—
352	C₂²⋊C₄	—	109	4	no	—
353	C₂²⋊C₄	—	109	4	no	—
354	C₂²⋊C₄	—	109	4	no	—
355	C₂²⋊C₄	—	109	4	no	—
356	C₂²⋊C₄	—	109	4	no	—
357	C₂²⋊C₄	—	109	4	no	—
358	C₂²⋊C₄	—	109	4	no	—
359	C₂²⋊C₄	—	109	4	no	—
360	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
361	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
362	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
363	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
364	C₄²	—	109	4	no	—
365	C₄²	C₂²×D₁₀₉	1	4	yes	complemented
366	C₂³	—	2	2	no	—
367	C₂³	—	2	2	no	—
368	C₂³	—	2	2	no	—
369	C₂³	—	2	2	no	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	2	no	—
372	C₂³	—	109	4	no	—
373	C₂³	—	109	2	no	—
374	C₂³	—	109	2	no	—
375	C₂³	—	109	2	no	—
376	C₂³	—	436	4	no	—
377	C₂³	—	436	4	no	—
378	C₂³	—	436	16	no	—
379	C₂³	—	218	8	no	—
380	C₂³	—	218	2	no	—
381	C₂³	—	218	2	no	—
382	C₂³	—	218	2	no	—
383	C₂³	—	218	2	no	—
384	C₂³	—	218	2	no	—
385	C₂³	—	218	2	no	—
386	C₂³	—	218	2	no	—
387	C₂³	—	218	2	no	—
388	C₂³	C₂³×D₁₀₉	1	2	yes	—
389	C₂³	C₂³×D₁₀₉	1	2	yes	—
390	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
391	Q₈	—	4	4	no	—
392	D₄	—	4	4	no	—
393	D₄	—	436	4	no	—
394	D₄	—	436	4	no	—
395	D₄	—	436	4	no	—
396	D₄	—	436	4	no	—
397	D₄	—	218	8	no	—
398	D₄	—	218	8	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	2	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	2	no	—
404	C₂×C₄	—	109	2	no	—
405	C₂×C₄	—	109	2	no	—
406	C₂×C₄	—	109	2	no	—
407	C₂×C₄	—	109	2	no	—
408	C₂×C₄	—	109	2	no	—
409	C₂×C₄	—	436	4	no	—
410	C₂×C₄	—	436	4	no	—
411	C₂×C₄	—	436	4	no	—
412	C₂×C₄	—	436	4	no	—
413	C₂×C₄	—	218	2	no	—
414	C₂×C₄	—	218	2	no	—
415	C₂×C₄	—	218	2	no	—
416	C₂×C₄	—	218	2	no	—
417	C₂×C₄	—	218	2	no	—
418	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
419	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
420	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
421	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
422	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	2	no	—
425	C₂²	—	2	2	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	2	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	4	4	no	—
430	C₂²	—	4	4	no	—
431	C₂²	—	109	4	no	—
432	C₂²	—	436	16	no	—
433	C₂²	—	436	4	no	—
434	C₂²	—	436	4	no	—
435	C₂²	—	218	2	no	—
436	C₂²	—	218	2	no	—
437	C₂²	—	218	2	no	—
438	C₂²	—	218	2	no	—
439	C₂²	—	218	2	no	—
440	C₂²	—	218	2	no	—
441	C₂²	—	218	2	no	—
442	C₂²	—	218	2	no	—
443	C₂²	—	218	8	no	—
444	C₂²	#41×D₁₀₉	1	2	yes	—
445	C₂²	#16812744	1	2	yes	—
446	C₂²	#16812745	1	2	yes	—
447	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
448	C₄	—	2	2	no	—
449	C₄	—	2	2	no	—
450	C₄	—	2	2	no	—
451	C₄	—	2	2	no	—
452	C₄	—	2	2	no	—
453	C₄	—	218	2	no	—
454	C₄	—	218	2	no	—
455	C₄	—	218	2	no	—
456	C₄	—	218	2	no	—
457	C₄	—	218	2	no	—
458	C₂	—	4	2	no	—
459	C₂	—	4	2	no	—
460	C₂	—	436	2	no	—
461	C₂	—	436	2	no	—
462	C₂	—	218	4	no	—
463	C₂	#20295610	1	4	yes	direct product factor
464	C₂	C₂×#41×D₁₀₉	1	1	yes	characteristic
465	C₂	C₂×#16812744	1	1	yes	characteristic
466	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×#124×C₁₀₉	C₂
socle	C₂³×C₁₀₉	C₂⁴
hypercenter	C₂×#124	D₁₀₉
hypocenter	C₁₀₉	C₂×#538
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂³ ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #20295610	C₂	direct (N × H)
2. C₂×#16812693	C₂	semidirect
3. C₂×#16812692	C₂	semidirect
4. C₂×#16812688	C₂	semidirect
5. C₂×#16812683	C₂	semidirect
6. C₂×#16812681	C₂	semidirect
7. C₂×#16812675	C₂	semidirect
8. C₂×#16812672	C₂	semidirect
9. C₂×#16812672	C₂	semidirect
10. C₂×#16812671	C₂	semidirect
11. C₂×#16812669	C₂	semidirect
12. C₂×#16812667	C₂	semidirect
13. C₂×C₄×D₄₃₆	C₂	semidirect
14. C₂×#16812659	C₂	semidirect
15. C₂×C₂²⋊C₄×D₁₀₉	C₂	semidirect
16. C₂×#124×C₁₀₉	C₂	semidirect
23. C₂²×C₂²⋊D₁₀₉	C₂²	semidirect
24. C₂³×Dic₁₀₉	C₂²	semidirect
27. C₂²×C₄×D₁₀₉	C₂²	semidirect
28. #16812693	C₂²	semidirect
29. #16812692	C₂²	semidirect
30. #16812688	C₂²	semidirect
31. #16812683	C₂²	semidirect
32. #16812681	C₂²	semidirect
33. #16812675	C₂²	semidirect
34. #16812672	C₂²	semidirect
35. #16812672	C₂²	semidirect
36. #16812671	C₂²	semidirect
37. #16812669	C₂²	semidirect
38. #16812667	C₂²	semidirect
39. C₄×D₄₃₆	C₂²	semidirect
40. #16812659	C₂²	semidirect
41. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
42. C₂×C₄⋊Dic₁₀₉	C₂²	semidirect
43. C₂×#13182008	C₂²	semidirect
44. C₂×#13182008	C₂²	semidirect
45. C₂×#13182008	C₂²	semidirect
46. C₂×#13182008	C₂²	semidirect
55. C₂×#13182006	C₂²	semidirect
56. C₂×#13182006	C₂²	semidirect
57. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
58. C₂×C₄×Dic₁₀₉	C₂²	semidirect
59. C₂²×Q₈×C₁₀₉	C₂²	semidirect
61. #124×C₁₀₉	C₂²	semidirect
64. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
65. C₂×C₂²⋊C₄×C₁₀₉	C₂²	semidirect
66. C₂×C₄²×C₁₀₉	C₂²	semidirect
85. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
92. C₂²×Dic₁₀₉	C₂³	semidirect
98. C₂×C₄×D₁₀₉	C₂³	semidirect
99. C₄⋊Dic₁₀₉	C₂³	semidirect
100. C₄⋊Dic₁₀₉	C₂³	semidirect
101. #13182008	C₂³	semidirect
102. #13182008	C₂³	semidirect
103. #13182008	C₂³	semidirect
104. #13182008	C₂³	semidirect
113. #13182006	C₂³	semidirect
114. #13182006	C₂³	semidirect
115. C₄×Dic₁₀₉	C₂³	semidirect
119. C₂×Q₈×C₁₀₉	C₂³	semidirect
129. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
130. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
131. C₄²×C₁₀₉	C₂³	semidirect
226. C₂×C₁₀₉	#538	semidirect
234. C₁₀₉	C₂×#538	semidirect
239. C₂×#124	D₁₀₉	semidirect
252. C₂²×Q₈	C₂×D₁₀₉	semidirect
267. #124	C₂×D₁₀₉	semidirect
297. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
298. C₂×C₂²⋊C₄	C₂×D₁₀₉	semidirect
300. C₂×C₄²	C₂×D₁₀₉	semidirect
311. C₂×Q₈	C₂²×D₁₀₉	semidirect
362. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
363. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
365. C₄²	C₂²×D₁₀₉	semidirect
463. C₂	#20295610	direct (N × H)

Maximal subgroups

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

subgroup	index
#20295610	2
C₂×#16812693	2
C₂×#16812692	2
C₂×#16812688	2
C₂×#16812683	2
C₂×#16812681	2
C₂×#16812675	2
C₂×#16812672	2
C₂×#16812671	2
C₂×#16812669	2
C₂×#16812667	2
C₂×C₄×D₄₃₆	2
C₂×#16812659	2
C₂×C₂²⋊C₄×D₁₀₉	2
C₂×#124×C₁₀₉	2
C₂×#538	109

Subgroups — C2×#20295610 #28376369

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20295610 #28376369