Subgroups — #28376665

Order 13952.

1900 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812745	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295672	C₂	1	4	yes	maximal, complemented
4	#20295667	C₂	1	4	yes	maximal, complemented
5	D₄×D₄₃₆	C₂	1	4	yes	maximal, complemented
6	#20295660	C₂	1	4	yes	maximal, complemented
7	#20295595	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295590	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295587	C₂	1	4	yes	maximal, complemented
10	#20295537	C₂	1	1	yes	maximal, characteristic, complemented
11	#120×D₁₀₉	C₂	1	4	yes	maximal, complemented
12	#20295500	C₂	1	1	yes	maximal, characteristic, complemented
13	#525×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂⁴×D₁₀₉	C₂²	1	2	yes	complemented
15	#16812745	—	2	8	no	—
16	C₂×#13182030	C₂²	1	2	yes	complemented
17	C₂×#13182029	C₂²	1	1	yes	characteristic, complemented
18	C₂×#13182029	C₂²	1	2	yes	complemented
19	C₂²×D₄₃₆	—	2	8	no	—
20	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic
21	C₂²×D₄₃₆	C₂²	1	2	yes	complemented
22	C₂²×D₄₃₆	C₂²	1	1	yes	characteristic, complemented
23	C₂²×D₄₃₆	C₂²	1	4	yes	complemented
24	C₂²×D₄₃₆	C₂²	1	4	yes	complemented
25	C₂²×D₄₃₆	C₂²	1	2	yes	complemented
26	C₂×D₄×D₁₀₉	C₂²	1	4	yes	complemented
27	C₂×D₄×D₁₀₉	C₂²	1	8	yes	complemented
28	C₂×D₄×D₁₀₉	C₂²	1	2	yes	complemented
29	#16812696	C₂²	1	2	yes	complemented
30	#16812695	C₂²	1	2	yes	complemented
31	#16812688	C₂²	1	2	yes	complemented
32	#16812687	C₂²	1	2	yes	complemented
33	#16812684	C₂²	1	8	yes	complemented
34	#16812682	C₂²	1	2	yes	complemented
35	#16812678	C₂²	1	2	yes	complemented
36	#16812674	C₂²	1	8	yes	complemented
37	#16812674	C₂²	1	4	yes	complemented
38	#16812673	C₂²	1	2	yes	complemented
39	#16812673	C₂²	1	4	yes	complemented
40	#16812672	C₂²	1	8	yes	complemented
41	#16812669	C₂²	1	4	yes	complemented
42	#16812669	C₂²	1	8	yes	complemented
43	#16812669	C₂²	1	8	yes	complemented
44	#16812668	C₂²	1	2	yes	complemented
45	#16812667	C₂²	1	4	yes	complemented
46	#16812667	C₂²	1	8	yes	complemented
47	C₄×D₄₃₆	C₂²	1	4	yes	complemented
48	#16812661	C₂²	1	4	yes	complemented
49	C₄×C₂²⋊D₁₀₉	C₂²	1	4	yes	complemented
50	#16812658	C₂²	1	1	yes	characteristic, complemented
51	C₂²⋊C₄×D₁₀₉	C₂²	1	4	yes	complemented
52	C₂²⋊C₄×D₁₀₉	C₂²	1	8	yes	complemented
53	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
54	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#123×C₁₀₉	C₂²	1	2	yes	complemented
56	C₂²⋊Q₈×C₁₀₉	C₂²	1	2	yes	complemented
57	#121×C₁₀₉	C₂²	1	2	yes	complemented
58	#120×C₁₀₉	C₂²	1	2	yes	complemented
59	C₄×D₄×C₁₀₉	C₂²	1	4	yes	complemented
60	#117×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
61	C₂³×D₁₀₉	—	2	8	no	—
62	C₂³×D₁₀₉	—	4	8	no	—
63	C₂³×D₁₀₉	C₂³	1	8	yes	—
64	C₂³×D₁₀₉	C₂³	1	2	yes	—
65	C₂³×D₁₀₉	C₂³	1	2	yes	—
66	C₂³×D₁₀₉	C₂³	1	4	yes	complemented
67	#13182030	—	2	8	no	—
68	#13182029	—	2	4	no	—
69	#13182029	—	2	8	no	—
70	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic, complemented
71	C₂×D₄₃₆	—	2	8	no	—
72	C₂×D₄₃₆	—	2	4	no	—
73	C₂×D₄₃₆	—	2	16	no	—
74	C₂×D₄₃₆	—	2	16	no	—
75	C₂×D₄₃₆	—	2	8	no	—
76	C₂×D₄₃₆	D₄	1	8	yes	complemented
77	C₂×D₄₃₆	C₂³	1	8	yes	—
78	C₂×D₄₃₆	C₂³	1	2	yes	—
79	C₂×D₄₃₆	C₂³	1	2	yes	—
80	C₂×D₄₃₆	C₂³	1	2	yes	—
81	C₂×D₄₃₆	C₂³	1	2	yes	—
82	C₂×D₄₃₆	C₂³	1	4	yes	complemented
83	C₂×D₄₃₆	C₂³	1	1	yes	characteristic, complemented
84	C₂×D₄₃₆	C₂³	1	8	yes	complemented
85	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	—
86	C₂×C₂²⋊D₁₀₉	C₂³	1	2	yes	—
87	C₂×C₂²⋊D₁₀₉	C₂³	1	4	yes	complemented
88	C₂×C₂²⋊D₁₀₉	C₂³	1	8	yes	complemented
89	D₄×D₁₀₉	—	2	16	no	—
90	D₄×D₁₀₉	—	2	8	no	—
91	D₄×D₁₀₉	—	4	16	no	—
92	C₂×C₄×D₁₀₉	C₂³	1	4	yes	complemented
93	C₂×C₄×D₁₀₉	C₂³	1	2	yes	complemented
94	C₂×C₄×D₁₀₉	C₂³	1	8	yes	complemented
95	C₄⋊Dic₁₀₉	C₂³	1	2	yes	complemented
96	#13182008	C₂³	1	4	yes	complemented
97	C₂²⋊Dic₁₀₉	C₂³	1	2	yes	—
98	C₂²⋊Dic₁₀₉	C₂³	1	4	yes	—
99	#13182006	C₂³	1	8	yes	complemented
100	#13182006	C₂³	1	8	yes	complemented
101	#13182006	C₂³	1	8	yes	complemented
102	#13182006	C₂³	1	4	yes	complemented
103	C₄×Dic₁₀₉	C₂³	1	2	yes	complemented
104	C₂⁴×C₁₀₉	C₂³	1	1	yes	characteristic
105	#41×C₁₀₉	—	2	4	no	—
106	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
107	C₂×D₄×C₁₀₉	C₂³	1	4	yes	—
108	C₂×D₄×C₁₀₉	C₂³	1	2	yes	—
109	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic, complemented
110	C₂²×C₄×C₁₀₉	—	2	4	no	—
111	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
112	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
113	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
114	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
115	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	complemented
116	C₄⋊C₄×C₁₀₉	C₂³	1	2	yes	complemented
117	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
118	C₂²⋊C₄×C₁₀₉	C₂³	1	2	yes	—
119	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	—
120	C₂²⋊C₄×C₁₀₉	C₂³	1	4	yes	complemented
121	C₄²×C₁₀₉	C₂³	1	2	yes	complemented
122	C₂³×C₁₀₉	—	2	2	no	—
123	C₂³×C₁₀₉	—	2	4	no	—
124	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
125	C₂³×C₁₀₉	C₂⁴	1	1	yes	characteristic
126	C₂³×C₁₀₉	C₂⁴	1	2	yes	—
127	C₂²×D₁₀₉	—	2	4	no	—
128	C₂²×D₁₀₉	—	2	16	no	—
129	C₂²×D₁₀₉	—	2	16	no	—
130	C₂²×D₁₀₉	—	2	8	no	—
131	C₂²×D₁₀₉	—	4	4	no	—
132	C₂²×D₁₀₉	—	4	8	no	—
133	C₂²×D₁₀₉	—	4	4	no	—
134	C₂²×D₁₀₉	—	4	8	no	—
135	C₂²×D₁₀₉	—	4	32	no	—
136	C₂²×D₁₀₉	C₂×D₄	1	8	yes	complemented
137	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
138	C₂²×D₁₀₉	C₂⁴	1	2	yes	—
139	C₂²×D₁₀₉	C₂⁴	1	4	yes	—
140	Q₈×C₁₀₉	—	2	2	no	—
141	D₄×C₁₀₉	—	2	4	no	—
142	D₄×C₁₀₉	—	2	8	no	—
143	D₄×C₁₀₉	—	2	2	no	—
144	C₂×C₄×C₁₀₉	—	2	2	no	—
145	C₂×C₄×C₁₀₉	—	2	4	no	—
146	C₂×C₄×C₁₀₉	—	2	4	no	—
147	C₂×C₄×C₁₀₉	—	2	4	no	—
148	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	complemented
149	C₂×C₄×C₁₀₉	C₂⁴	1	4	yes	—
150	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
151	C₂×C₄×C₁₀₉	C₂⁴	1	2	yes	—
152	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
153	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
154	C₂²⋊D₁₀₉	—	2	4	no	—
155	C₂²⋊D₁₀₉	—	2	8	no	—
156	C₂²⋊D₁₀₉	—	2	8	no	—
157	C₂²⋊D₁₀₉	—	4	8	no	—
158	C₂×Dic₁₀₉	C₂⁴	1	4	yes	—
159	C₂×Dic₁₀₉	C₂⁴	1	2	yes	—
160	D₄₃₆	—	2	8	no	—
161	D₄₃₆	—	2	16	no	—
162	D₄₃₆	—	2	4	no	—
163	D₄₃₆	—	2	4	no	—
164	D₄₃₆	—	2	2	no	—
165	D₄₃₆	—	4	8	no	—
166	D₄₃₆	C₂×D₄	1	8	yes	complemented
167	D₄₃₆	C₂×D₄	1	8	yes	complemented
168	C₄×D₁₀₉	—	2	8	no	—
169	C₄×D₁₀₉	—	2	4	no	—
170	C₄×D₁₀₉	—	4	8	no	—
171	Dic₂₁₈	—	2	2	no	—
172	C₂²×C₁₀₉	—	2	8	no	—
173	C₂²×C₁₀₉	—	2	2	no	—
174	C₂²×C₁₀₉	—	2	2	no	—
175	C₂²×C₁₀₉	—	2	2	no	—
176	C₂²×C₁₀₉	—	2	2	no	—
177	C₂²×C₁₀₉	—	2	4	no	—
178	C₂²×C₁₀₉	C₂²×D₄	1	2	yes	—
179	C₂²×C₁₀₉	H(5,𝔽₂)	1	4	yes	complemented
180	C₂²×C₁₀₉	C₂⁵	1	1	yes	characteristic
181	C₂×D₁₀₉	—	2	8	no	—
182	C₂×D₁₀₉	—	2	2	no	—
183	C₂×D₁₀₉	—	2	4	no	—
184	C₂×D₁₀₉	—	2	4	no	—
185	C₂×D₁₀₉	—	2	8	no	—
186	C₂×D₁₀₉	—	4	8	no	—
187	C₂×D₁₀₉	—	4	32	no	—
188	C₂×D₁₀₉	—	4	8	no	—
189	C₂×D₁₀₉	—	8	8	no	—
190	C₂×D₁₀₉	C₂²×D₄	1	8	yes	—
191	C₄×C₁₀₉	—	2	2	no	—
192	C₄×C₁₀₉	—	2	2	no	—
193	C₄×C₁₀₉	—	2	4	no	—
194	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
195	C₄×C₁₀₉	C₂²×D₄	1	2	yes	—
196	Dic₁₀₉	—	2	2	no	—
197	Dic₁₀₉	—	2	4	no	—
198	C₂×C₁₀₉	—	2	2	no	—
199	C₂×C₁₀₉	—	2	4	no	—
200	C₂×C₁₀₉	—	4	2	no	—
201	C₂×C₁₀₉	C₂³×D₄	1	1	yes	characteristic
202	C₂×C₁₀₉	C₂×H(5,𝔽₂)	1	2	yes	—
203	D₁₀₉	—	2	8	no	—
204	D₁₀₉	—	4	4	no	—
205	D₁₀₉	—	4	2	no	—
206	#3484	—	109	1	no	maximal, Sylow
207	C₁₀₉	#3484	1	1	yes	characteristic, complemented, Sylow
208	C₂×H(5,𝔽₂)	—	109	1	no	—
209	C₂³×D₄	—	109	1	no	—
210	#546	—	109	4	no	—
211	#546	—	109	4	no	—
212	D₄²	—	109	4	no	—
213	D₄²	—	109	4	no	—
214	#535	—	109	1	no	—
215	#535	—	109	1	no	—
216	#534	—	109	4	no	—
217	#525	—	109	1	no	—
218	#525	D₁₀₉	1	1	yes	characteristic, complemented
219	C₂×#120	—	109	4	no	—
220	#518	—	109	1	no	—
221	C₂⁵	—	109	2	no	—
222	H(5,𝔽₂)	—	218	8	no	—
223	C₂×#41	—	109	2	no	—
224	C₂×#41	—	109	1	no	—
225	C₂×#41	—	109	2	no	—
226	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic, complemented
227	C₂²×D₄	—	109	8	no	—
228	C₂²×D₄	—	109	4	no	—
229	C₂²×D₄	—	109	2	no	—
230	C₂²×D₄	—	109	1	no	—
231	C₂²×D₄	—	109	2	no	—
232	C₂²×D₄	—	109	1	no	—
233	C₂²×D₄	—	109	4	no	—
234	C₂²×D₄	—	109	4	no	—
235	C₂²×D₄	—	109	2	no	—
236	C₂²×D₄	—	218	8	no	—
237	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic, complemented
238	#127	—	109	2	no	—
239	#127	—	109	2	no	—
240	#124	—	109	2	no	—
241	#124	—	109	2	no	—
242	#123	—	109	2	no	—
243	#123	—	109	8	no	—
244	#123	C₂×D₁₀₉	1	2	yes	complemented
245	C₂²⋊Q₈	—	109	2	no	—
246	C₂²⋊Q₈	C₂×D₁₀₉	1	2	yes	complemented
247	#121	—	109	2	no	—
248	#121	—	109	4	no	—
249	#121	—	109	8	no	—
250	#121	—	109	4	no	—
251	#121	—	109	8	no	—
252	#121	C₂×D₁₀₉	1	2	yes	complemented
253	#120	—	109	2	no	—
254	#120	—	109	4	no	—
255	#120	—	109	8	no	—
256	#120	—	109	4	no	—
257	#120	—	109	8	no	—
258	#120	—	109	8	no	—
259	#120	C₂×D₁₀₉	1	2	yes	complemented
260	C₄×D₄	—	109	4	no	—
261	C₄×D₄	—	109	4	no	—
262	C₄×D₄	—	109	4	no	—
263	C₄×D₄	C₂×D₁₀₉	1	4	yes	complemented
264	#117	—	109	1	no	—
265	#117	C₂×D₁₀₉	1	1	yes	characteristic, complemented
266	C₂×C₂²⋊C₄	—	109	4	no	—
267	C₂×C₂²⋊C₄	—	109	8	no	—
268	C₂⁴	—	109	2	no	—
269	C₂⁴	—	109	2	no	—
270	C₂⁴	—	109	4	no	—
271	C₂⁴	—	109	8	no	—
272	C₂⁴	—	436	8	no	—
273	C₂⁴	—	218	8	no	—
274	C₂⁴	C₂²×D₁₀₉	1	1	yes	characteristic
275	#41	—	2	4	no	—
276	#41	—	218	8	no	—
277	#41	—	218	8	no	—
278	#41	—	218	4	no	—
279	C₂×Q₈	—	109	1	no	—
280	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
281	C₂×D₄	—	109	8	no	—
282	C₂×D₄	—	109	4	no	—
283	C₂×D₄	—	109	2	no	—
284	C₂×D₄	—	109	4	no	—
285	C₂×D₄	—	109	8	no	—
286	C₂×D₄	—	109	2	no	—
287	C₂×D₄	—	109	2	no	—
288	C₂×D₄	—	109	8	no	—
289	C₂×D₄	—	109	8	no	—
290	C₂×D₄	—	109	2	no	—
291	C₂×D₄	—	109	2	no	—
292	C₂×D₄	—	109	4	no	—
293	C₂×D₄	—	109	1	no	—
294	C₂×D₄	—	436	16	no	—
295	C₂×D₄	—	218	16	no	—
296	C₂×D₄	—	218	16	no	—
297	C₂×D₄	—	218	8	no	—
298	C₂×D₄	—	218	8	no	—
299	C₂×D₄	—	218	8	no	—
300	C₂×D₄	—	218	16	no	—
301	C₂×D₄	—	218	4	no	—
302	C₂×D₄	C₂²×D₁₀₉	1	4	yes	—
303	C₂×D₄	C₂²×D₁₀₉	1	2	yes	—
304	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic, complemented
305	C₂²×C₄	—	2	4	no	—
306	C₂²×C₄	—	109	8	no	—
307	C₂²×C₄	—	109	4	no	—
308	C₂²×C₄	—	109	2	no	—
309	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
310	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
311	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
312	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
313	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	complemented
314	C₄⋊C₄	—	109	4	no	—
315	C₄⋊C₄	—	109	2	no	—
316	C₄⋊C₄	C₂²×D₁₀₉	1	2	yes	complemented
317	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
318	C₂²⋊C₄	—	109	2	no	—
319	C₂²⋊C₄	—	109	4	no	—
320	C₂²⋊C₄	—	109	8	no	—
321	C₂²⋊C₄	—	109	8	no	—
322	C₂²⋊C₄	—	109	8	no	—
323	C₂²⋊C₄	—	109	4	no	—
324	C₂²⋊C₄	C₂²×D₁₀₉	1	2	yes	—
325	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	—
326	C₂²⋊C₄	C₂²×D₁₀₉	1	4	yes	complemented
327	C₄²	—	109	2	no	—
328	C₄²	C₂²×D₁₀₉	1	2	yes	complemented
329	C₂³	—	2	2	no	—
330	C₂³	—	2	4	no	—
331	C₂³	—	109	8	no	—
332	C₂³	—	109	4	no	—
333	C₂³	—	109	4	no	—
334	C₂³	—	109	2	no	—
335	C₂³	—	436	8	no	—
336	C₂³	—	436	32	no	—
337	C₂³	—	436	4	no	—
338	C₂³	—	436	8	no	—
339	C₂³	—	436	4	no	—
340	C₂³	—	218	8	no	—
341	C₂³	—	218	16	no	—
342	C₂³	—	218	4	no	—
343	C₂³	—	218	16	no	—
344	C₂³	C₂³×D₁₀₉	1	2	yes	—
345	C₂³	C₂³×D₁₀₉	1	1	yes	characteristic
346	C₂³	C₂³×D₁₀₉	1	2	yes	—
347	Q₈	—	2	2	no	—
348	Q₈	—	218	2	no	—
349	D₄	—	2	4	no	—
350	D₄	—	2	8	no	—
351	D₄	—	2	2	no	—
352	D₄	—	109	8	no	—
353	D₄	—	109	8	no	—
354	D₄	—	436	8	no	—
355	D₄	—	436	8	no	—
356	D₄	—	218	16	no	—
357	D₄	—	218	8	no	—
358	D₄	—	218	8	no	—
359	D₄	—	218	4	no	—
360	D₄	—	218	8	no	—
361	D₄	—	218	4	no	—
362	D₄	—	218	4	no	—
363	D₄	—	218	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	4	no	—
368	C₂×C₄	—	109	4	no	—
369	C₂×C₄	—	109	2	no	—
370	C₂×C₄	—	436	8	no	—
371	C₂×C₄	—	218	8	no	—
372	C₂×C₄	—	218	4	no	—
373	C₂×C₄	D₄×D₁₀₉	1	4	yes	complemented
374	C₂×C₄	C₂³×D₁₀₉	1	4	yes	—
375	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
376	C₂×C₄	C₂³×D₁₀₉	1	2	yes	—
377	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
378	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
379	C₂²	—	2	8	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	2	4	no	—
385	C₂²	—	872	8	no	—
386	C₂²	—	109	8	no	—
387	C₂²	—	436	8	no	—
388	C₂²	—	436	32	no	—
389	C₂²	—	436	8	no	—
390	C₂²	—	218	2	no	—
391	C₂²	—	218	4	no	—
392	C₂²	—	218	4	no	—
393	C₂²	—	218	8	no	—
394	C₂²	—	218	8	no	—
395	C₂²	C₂×D₄×D₁₀₉	1	2	yes	—
396	C₂²	#16812745	1	4	yes	complemented
397	C₂²	C₂⁴×D₁₀₉	1	1	yes	characteristic
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	4	no	—
401	C₄	—	218	2	no	—
402	C₄	—	218	4	no	—
403	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
404	C₄	C₂×D₄×D₁₀₉	1	2	yes	—
405	C₂	—	2	2	no	—
406	C₂	—	2	4	no	—
407	C₂	—	4	2	no	—
408	C₂	—	436	2	no	—
409	C₂	—	436	4	no	—
410	C₂	—	218	8	no	—
411	C₂	C₂²×D₄×D₁₀₉	1	1	yes	characteristic
412	C₂	C₂×#16812745	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₀₉
commutator, socle, cosocle	C₂²×C₁₀₉	C₂⁵
Fitting	#525×C₁₀₉	C₂
hypercenter	#525	D₁₀₉
hypocenter	C₁₀₉	#3484
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16812745	C₂	semidirect
2. C₂³×D₄₃₆	C₂	semidirect
3. #20295672	C₂	semidirect
4. #20295667	C₂	semidirect
5. D₄×D₄₃₆	C₂	semidirect
6. #20295660	C₂	semidirect
7. #20295595	C₂	semidirect
8. #20295590	C₂	semidirect
9. #20295587	C₂	semidirect
10. #20295537	C₂	semidirect
11. #120×D₁₀₉	C₂	semidirect
12. #20295500	C₂	semidirect
13. #525×C₁₀₉	C₂	semidirect
14. C₂⁴×D₁₀₉	C₂²	semidirect
16. C₂×#13182030	C₂²	semidirect
17. C₂×#13182029	C₂²	semidirect
18. C₂×#13182029	C₂²	semidirect
21. C₂²×D₄₃₆	C₂²	semidirect
22. C₂²×D₄₃₆	C₂²	semidirect
23. C₂²×D₄₃₆	C₂²	semidirect
24. C₂²×D₄₃₆	C₂²	semidirect
25. C₂²×D₄₃₆	C₂²	semidirect
26. C₂×D₄×D₁₀₉	C₂²	semidirect
27. C₂×D₄×D₁₀₉	C₂²	semidirect
28. C₂×D₄×D₁₀₉	C₂²	semidirect
29. #16812696	C₂²	semidirect
30. #16812695	C₂²	semidirect
31. #16812688	C₂²	semidirect
32. #16812687	C₂²	semidirect
33. #16812684	C₂²	semidirect
34. #16812682	C₂²	semidirect
35. #16812678	C₂²	semidirect
36. #16812674	C₂²	semidirect
37. #16812674	C₂²	semidirect
38. #16812673	C₂²	semidirect
39. #16812673	C₂²	semidirect
40. #16812672	C₂²	semidirect
41. #16812669	C₂²	semidirect
42. #16812669	C₂²	semidirect
43. #16812669	C₂²	semidirect
44. #16812668	C₂²	semidirect
45. #16812667	C₂²	semidirect
46. #16812667	C₂²	semidirect
47. C₄×D₄₃₆	C₂²	semidirect
48. #16812661	C₂²	semidirect
49. C₄×C₂²⋊D₁₀₉	C₂²	semidirect
50. #16812658	C₂²	semidirect
51. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
52. C₂²⋊C₄×D₁₀₉	C₂²	semidirect
53. C₂×#41×C₁₀₉	C₂²	semidirect
54. C₂³×C₄×C₁₀₉	C₂²	semidirect
55. #123×C₁₀₉	C₂²	semidirect
56. C₂²⋊Q₈×C₁₀₉	C₂²	semidirect
57. #121×C₁₀₉	C₂²	semidirect
58. #120×C₁₀₉	C₂²	semidirect
59. C₄×D₄×C₁₀₉	C₂²	semidirect
60. #117×C₁₀₉	C₂²	semidirect
66. C₂³×D₁₀₉	C₂³	semidirect
70. C₂×Dic₂₁₈	C₂³	semidirect
76. C₂×D₄₃₆	D₄	semidirect
82. C₂×D₄₃₆	C₂³	semidirect
83. C₂×D₄₃₆	C₂³	semidirect
84. C₂×D₄₃₆	C₂³	semidirect
87. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
88. C₂×C₂²⋊D₁₀₉	C₂³	semidirect
92. C₂×C₄×D₁₀₉	C₂³	semidirect
93. C₂×C₄×D₁₀₉	C₂³	semidirect
94. C₂×C₄×D₁₀₉	C₂³	semidirect
95. C₄⋊Dic₁₀₉	C₂³	semidirect
96. #13182008	C₂³	semidirect
99. #13182006	C₂³	semidirect
100. #13182006	C₂³	semidirect
101. #13182006	C₂³	semidirect
102. #13182006	C₂³	semidirect
103. C₄×Dic₁₀₉	C₂³	semidirect
106. C₂×Q₈×C₁₀₉	C₂³	semidirect
109. C₂×D₄×C₁₀₉	C₂³	semidirect
115. C₂²×C₄×C₁₀₉	C₂³	semidirect
116. C₄⋊C₄×C₁₀₉	C₂³	semidirect
117. C₄⋊C₄×C₁₀₉	C₂³	semidirect
120. C₂²⋊C₄×C₁₀₉	C₂³	semidirect
121. C₄²×C₁₀₉	C₂³	semidirect
136. C₂²×D₁₀₉	C₂×D₄	semidirect
148. C₂×C₄×C₁₀₉	C₂×D₄	semidirect
166. D₄₃₆	C₂×D₄	semidirect
167. D₄₃₆	C₂×D₄	semidirect
179. C₂²×C₁₀₉	H(5,𝔽₂)	semidirect
207. C₁₀₉	#3484	semidirect
218. #525	D₁₀₉	semidirect
226. C₂×#41	C₂×D₁₀₉	semidirect
237. C₂³×C₄	C₂×D₁₀₉	semidirect
244. #123	C₂×D₁₀₉	semidirect
246. C₂²⋊Q₈	C₂×D₁₀₉	semidirect
252. #121	C₂×D₁₀₉	semidirect
259. #120	C₂×D₁₀₉	semidirect
263. C₄×D₄	C₂×D₁₀₉	semidirect
265. #117	C₂×D₁₀₉	semidirect
280. C₂×Q₈	C₂²×D₁₀₉	semidirect
304. C₂×D₄	C₂²×D₁₀₉	semidirect
313. C₂²×C₄	C₂²×D₁₀₉	semidirect
316. C₄⋊C₄	C₂²×D₁₀₉	semidirect
317. C₄⋊C₄	C₂²×D₁₀₉	semidirect
326. C₂²⋊C₄	C₂²×D₁₀₉	semidirect
328. C₄²	C₂²×D₁₀₉	semidirect
373. C₂×C₄	D₄×D₁₀₉	semidirect
396. C₂²	#16812745	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812745	2
C₂³×D₄₃₆	2
#20295672	2
#20295667	2
D₄×D₄₃₆	2
#20295660	2
#20295595	2
#20295590	2
#20295587	2
#20295537	2
#120×D₁₀₉	2
#20295500	2
#525×C₁₀₉	2
#3484	109