Subgroups — #28376116

Order 13952.

380 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295761	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295416	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295413	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295408	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295248	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295245	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295241	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295238	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295221	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295218	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295157	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295126	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295124	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#13182014	C₂	1	1	yes	maximal, characteristic, complemented
15	#559×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182019	C₂²	1	1	yes	characteristic
17	C₂×#13182017	C₂²	1	1	yes	characteristic
18	C₂×#13182017	C₂²	1	1	yes	characteristic
19	C₂×#13182016	C₂²	1	1	yes	characteristic
20	C₂×#13182016	C₂²	1	1	yes	characteristic
21	C₂×#13182014	C₂²	1	1	yes	characteristic
22	C₂×#13182014	C₂²	1	1	yes	characteristic
23	C₂×#13182014	C₂²	1	1	yes	characteristic
24	C₂×#13182014	C₂²	1	1	yes	characteristic
25	#16812696	C₂²	1	1	yes	characteristic
26	#16812688	—	2	1	no	—
27	#16812687	C₂²	1	1	yes	characteristic
28	#16812687	C₂²	1	1	yes	characteristic
29	#16812684	—	2	1	no	—
30	#16812684	—	2	1	no	—
31	C₄×Dic₂₁₈	C₂²	1	1	yes	characteristic
32	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	#16812661	—	2	1	no	—
35	#16812632	C₂²	1	1	yes	characteristic
36	C₄⋊#11724784	C₂²	1	1	yes	characteristic
37	C₄⋊#11724784	C₂²	1	1	yes	characteristic
38	C₄⋊#11724784	C₂²	1	1	yes	characteristic
39	#16812623	C₂²	1	1	yes	characteristic
40	#16812622	C₂²	1	1	yes	characteristic
41	#16812620	C₂²	1	1	yes	characteristic
42	#16812620	C₂²	1	1	yes	characteristic
43	#16812619	C₂²	1	1	yes	characteristic
44	#16812619	C₂²	1	1	yes	characteristic
45	#16812610	C₂²	1	1	yes	characteristic
46	#16812610	C₂²	1	1	yes	characteristic
47	C₄×#11724784	C₂²	1	1	yes	characteristic
48	#127×C₁₀₉	C₂²	1	1	yes	characteristic
49	#124×C₁₀₉	—	2	1	no	—
50	#124×C₁₀₉	C₂²	1	1	yes	characteristic
51	#124×C₁₀₉	C₂²	1	1	yes	characteristic
52	#123×C₁₀₉	—	2	1	no	—
53	#123×C₁₀₉	—	2	1	no	—
54	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₁₀₉	—	2	1	no	—
57	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
58	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
60	C₂×D₄₃₆	—	2	1	no	—
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
63	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	C₂×C₄×D₁₀₉	—	2	1	no	—
66	#13182019	—	4	1	no	—
67	#13182017	—	2	2	no	—
68	#13182017	—	4	1	no	—
69	#13182016	—	2	2	no	—
70	#13182016	—	4	1	no	—
71	#13182014	—	2	2	no	—
72	#13182014	—	4	1	no	—
73	#13182014	—	4	1	no	—
74	#13182014	—	4	1	no	—
75	C₂×#11724784	C₂³	1	1	yes	characteristic
76	C₂×#11724784	C₂³	1	1	yes	characteristic
77	C₂×#11724784	C₂³	1	1	yes	characteristic
78	C₂×#11724784	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
80	#13182008	—	2	1	no	—
81	#13182006	—	2	1	no	—
82	#13182006	—	2	1	no	—
83	#13182006	—	2	1	no	—
84	#13182006	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	C₄×Dic₁₀₉	—	2	1	no	—
88	C₂×Q₈×C₁₀₉	D₄	1	1	yes	characteristic
89	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₁₀₉	—	2	1	no	—
91	C₂×D₄×C₁₀₉	—	2	1	no	—
92	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
93	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₁₀₉	—	2	1	no	—
95	C₂²×C₄×C₁₀₉	—	2	1	no	—
96	C₄⋊C₄×C₁₀₉	—	2	1	no	—
97	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
98	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
99	C₄⋊C₄×C₁₀₉	D₄	1	1	yes	characteristic
100	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
102	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
103	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	—	2	1	no	—
108	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
109	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
110	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
111	C₂³×C₁₀₉	—	2	1	no	—
112	C₂³×C₁₀₉	—	2	1	no	—
113	C₂²×D₁₀₉	—	2	1	no	—
114	C₂²×D₁₀₉	—	2	1	no	—
115	Q₈×C₁₀₉	—	2	1	no	—
116	Q₈×C₁₀₉	—	2	1	no	—
117	Q₈×C₁₀₉	—	4	1	no	—
118	D₄×C₁₀₉	—	2	1	no	—
119	D₄×C₁₀₉	—	2	1	no	—
120	D₄×C₁₀₉	—	2	1	no	—
121	D₄×C₁₀₉	—	2	1	no	—
122	D₄×C₁₀₉	—	4	2	no	—
123	D₄×C₁₀₉	—	8	1	no	—
124	C₂×C₄×C₁₀₉	—	2	1	no	—
125	C₂×C₄×C₁₀₉	—	2	1	no	—
126	C₂×C₄×C₁₀₉	—	8	1	no	—
127	C₂×C₄×C₁₀₉	—	8	1	no	—
128	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
135	C₂×Dic₁₀₉	—	2	1	no	—
136	C₂×Dic₁₀₉	—	2	1	no	—
137	D₄₃₆	—	2	1	no	—
138	D₄₃₆	—	2	1	no	—
139	D₄₃₆	—	2	1	no	—
140	D₄₃₆	—	2	1	no	—
141	D₄₃₆	—	4	2	no	—
142	D₄₃₆	—	8	1	no	—
143	C₄×D₁₀₉	—	8	1	no	—
144	C₄×D₁₀₉	—	8	1	no	—
145	Dic₂₁₈	—	2	1	no	—
146	Dic₂₁₈	—	2	1	no	—
147	#11724784	—	2	1	no	—
148	#11724784	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	#11724784	#41	1	2	yes	—
151	C₂²×C₁₀₉	—	4	1	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	4	1	no	—
156	C₂²×C₁₀₉	—	4	1	no	—
157	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×D₁₀₉	—	4	1	no	—
159	C₂×D₁₀₉	—	4	1	no	—
160	C₂×D₁₀₉	—	4	1	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	4	1	no	—
163	C₂×D₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	—	2	1	no	—
165	C₄×C₁₀₉	—	2	1	no	—
166	C₄×C₁₀₉	—	2	1	no	—
167	C₄×C₁₀₉	—	2	1	no	—
168	C₄×C₁₀₉	—	2	1	no	—
169	C₄×C₁₀₉	—	2	1	no	—
170	C₄×C₁₀₉	—	4	1	no	—
171	C₄×C₁₀₉	—	4	1	no	—
172	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
173	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
174	Dic₁₀₉	—	4	1	no	—
175	Dic₁₀₉	—	4	1	no	—
176	C₂×C₁₀₉	—	8	1	no	—
177	C₂×C₁₀₉	—	8	1	no	—
178	C₂×C₁₀₉	#550	1	1	yes	characteristic
179	C₂×C₁₀₉	#575	1	1	yes	characteristic
180	C₂×C₁₀₉	#576	1	1	yes	characteristic
181	D₁₀₉	—	8	1	no	—
182	D₁₀₉	—	8	1	no	—
183	#3403	—	109	1	no	maximal, Sylow
184	C₁₀₉	#3403	1	1	yes	characteristic, complemented, Sylow
185	#559	—	109	1	no	—
186	#559	D₁₀₉	1	1	yes	characteristic, complemented
187	#496	—	109	1	no	—
188	#496	—	109	1	no	—
189	#495	—	109	1	no	—
190	#464	—	109	1	no	—
191	#464	—	109	1	no	—
192	#463	—	109	1	no	—
193	#463	—	109	1	no	—
194	#459	—	109	1	no	—
195	#459	—	109	1	no	—
196	#446	—	109	1	no	—
197	#440	—	109	1	no	—
198	#440	—	109	1	no	—
199	C₄×D₈	—	109	1	no	—
200	C₂×Q₁₆	—	109	1	no	—
201	C₂×SD₁₆	—	109	1	no	—
202	C₂×SD₁₆	—	109	1	no	—
203	C₂×SD₁₆	—	109	1	no	—
204	C₂×SD₁₆	—	109	1	no	—
205	C₂×D₈	—	109	1	no	—
206	C₂×D₈	—	109	1	no	—
207	C₂×D₈	—	109	1	no	—
208	C₂×D₈	—	109	1	no	—
209	#127	—	109	1	no	—
210	#127	C₂×D₁₀₉	1	1	yes	characteristic
211	#124	—	2	1	no	—
212	#124	—	109	1	no	—
213	#124	—	109	1	no	—
214	#124	—	218	1	no	—
215	#124	C₂×D₁₀₉	1	1	yes	characteristic
216	#124	C₂×D₁₀₉	1	1	yes	characteristic
217	#123	—	2	1	no	—
218	#123	—	2	1	no	—
219	#123	—	218	1	no	—
220	#123	—	218	1	no	—
221	C₄×Q₈	—	109	1	no	—
222	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
223	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
224	C₄×D₄	—	2	1	no	—
225	C₄×D₄	—	109	1	no	—
226	C₄×D₄	—	109	1	no	—
227	C₄×D₄	—	218	1	no	—
228	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
229	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
230	#107	—	109	1	no	—
231	C₄⋊C₈	—	109	1	no	—
232	C₄⋊C₈	—	109	1	no	—
233	C₄⋊C₈	—	109	1	no	—
234	#103	—	109	1	no	—
235	#103	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#97	—	109	1	no	—
241	#97	—	109	1	no	—
242	C₄×C₈	—	109	1	no	—
243	C₂×Q₈	—	109	1	no	—
244	C₂×Q₈	C₂²⋊D₁₀₉	1	1	yes	characteristic
245	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	2	1	no	—
248	C₂×D₄	—	109	1	no	—
249	C₂×D₄	—	109	1	no	—
250	C₂×D₄	—	218	1	no	—
251	C₂×D₄	—	218	1	no	—
252	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	2	1	no	—
256	C₂²×C₄	—	218	1	no	—
257	C₂²×C₄	—	218	1	no	—
258	Q₁₆	—	436	1	no	—
259	SD₁₆	—	436	1	no	—
260	SD₁₆	—	436	1	no	—
261	SD₁₆	—	218	2	no	—
262	SD₁₆	—	218	2	no	—
263	D₈	—	436	1	no	—
264	D₈	—	436	1	no	—
265	D₈	—	436	1	no	—
266	D₈	—	218	2	no	—
267	C₂×C₈	—	109	1	no	—
268	C₂×C₈	—	109	1	no	—
269	C₂×C₈	—	109	1	no	—
270	C₂×C₈	—	109	1	no	—
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	109	1	no	—
273	C₄⋊C₄	—	218	1	no	—
274	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
275	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²⋊D₁₀₉	1	1	yes	characteristic
277	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	2	1	no	—
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	218	1	no	—
285	C₂²⋊C₄	—	218	1	no	—
286	C₂²⋊C₄	—	218	1	no	—
287	C₂²⋊C₄	—	218	1	no	—
288	C₂²⋊C₄	—	218	1	no	—
289	C₂²⋊C₄	—	218	1	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	218	1	no	—
292	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
293	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
294	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	2	1	no	—
297	C₂³	—	218	1	no	—
298	C₂³	—	218	1	no	—
299	Q₈	—	2	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	218	1	no	—
303	Q₈	—	218	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	4	2	no	—
309	D₄	—	8	1	no	—
310	D₄	—	872	1	no	—
311	D₄	—	436	2	no	—
312	D₄	—	218	1	no	—
313	D₄	—	218	1	no	—
314	D₄	—	218	1	no	—
315	D₄	—	218	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	8	1	no	—
320	C₂×C₄	—	872	1	no	—
321	C₂×C₄	—	872	1	no	—
322	C₂×C₄	—	218	1	no	—
323	C₂×C₄	—	218	1	no	—
324	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
325	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
326	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
327	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
328	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
329	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
331	C₈	—	109	2	no	—
332	C₈	—	218	1	no	—
333	C₈	—	218	1	no	—
334	C₈	—	218	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	436	1	no	—
342	C₂²	—	436	1	no	—
343	C₂²	—	436	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	436	1	no	—
347	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	1	no	—
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	436	1	no	—
357	C₄	—	436	1	no	—
358	C₄	#41×D₁₀₉	1	1	yes	characteristic
359	C₄	#16812745	1	1	yes	characteristic
360	C₂	—	8	1	no	—
361	C₂	—	8	1	no	—
362	C₂	—	872	1	no	—
363	C₂	—	872	1	no	—
364	C₂	#20295695	1	1	yes	characteristic
365	C₂	#20295824	1	1	yes	characteristic
366	C₂	#20295829	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#559×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#559	D₁₀₉
hypocenter	C₁₀₉	#3403
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #559

Complements

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

N	H ≅ G/N	type
1. #20295761	C₂	semidirect
2. #20295416	C₂	semidirect
3. #20295413	C₂	semidirect
4. #20295408	C₂	semidirect
5. #20295248	C₂	semidirect
6. #20295245	C₂	semidirect
7. #20295241	C₂	semidirect
8. #20295238	C₂	semidirect
9. #20295221	C₂	semidirect
10. #20295218	C₂	semidirect
11. #20295157	C₂	semidirect
12. #20295126	C₂	semidirect
13. #20295124	C₂	semidirect
14. C₄×#13182014	C₂	semidirect
15. #559×C₁₀₉	C₂	semidirect
184. C₁₀₉	#3403	semidirect
186. #559	D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295761	2
#20295416	2
#20295413	2
#20295408	2
#20295248	2
#20295245	2
#20295241	2
#20295238	2
#20295221	2
#20295218	2
#20295157	2
#20295126	2
#20295124	2
C₄×#13182014	2
#559×C₁₀₉	2
#3403	109