Subgroups — #28373277

Order 13952.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295632	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295498	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295353	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295352	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295323	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295222	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295221	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295161	C₂	1	1	yes	maximal, characteristic, complemented
9	#20295161	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295137	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295135	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295108	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295106	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295086	C₂	1	1	yes	maximal, characteristic, complemented
15	#478×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₁₀₉	C₂²	1	1	yes	characteristic
17	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
18	C₂×D₈₇₂	C₂²	1	1	yes	characteristic
19	C₂×#13182014	C₂²	1	1	yes	characteristic
20	C₂×#13182014	C₂²	1	1	yes	characteristic
21	#16812696	C₂²	1	1	yes	characteristic, complemented
22	#16812684	—	2	1	no	—
23	#16812684	—	2	1	no	—
24	#16812674	—	2	1	no	—
25	#16812674	—	2	1	no	—
26	#16812674	C₂²	1	1	yes	characteristic
27	#16812674	C₂²	1	1	yes	characteristic
28	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
30	#16812661	C₂²	1	1	yes	characteristic
31	#16812661	C₂²	1	1	yes	characteristic, complemented
32	#16812661	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₂²⋊D₁₀₉	—	2	1	no	—
34	#16812659	—	2	1	no	—
35	#16812657	C₂²	1	1	yes	characteristic
36	#16812656	C₂²	1	1	yes	characteristic
37	C₂×#13182006	—	2	1	no	—
38	C₂²⋊C₄×D₁₀₉	—	2	1	no	—
39	#16812632	C₂²	1	1	yes	characteristic, complemented
40	#16812630	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊#11724784	C₂²	1	1	yes	characteristic, complemented
42	#16812621	C₂²	1	1	yes	characteristic, complemented
43	#16812621	C₂²	1	1	yes	characteristic, complemented
44	#16812619	C₂²	1	1	yes	characteristic
45	#16812619	C₂²	1	1	yes	characteristic
46	#16812619	C₂²	1	1	yes	characteristic, complemented
47	#16812619	C₂²	1	1	yes	characteristic, complemented
48	#16812611	C₂²	1	1	yes	characteristic
49	#16812611	C₂²	1	1	yes	characteristic
50	#16812609	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₁₀₉	C₂²	1	1	yes	characteristic, complemented
52	#125×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₁₀₉	C₂²	1	1	yes	characteristic
54	#107×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂³×D₁₀₉	—	2	1	no	—
60	C₂×D₄₃₆	—	2	1	no	—
61	C₂×D₄₃₆	—	2	1	no	—
62	C₂×D₄₃₆	—	2	1	no	—
63	C₂×D₄₃₆	—	2	1	no	—
64	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
65	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
66	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
68	D₄×D₁₀₉	—	2	2	no	—
69	D₄×D₁₀₉	—	2	2	no	—
70	C₂²×Dic₁₀₉	—	2	1	no	—
71	C₂×C₄×D₁₀₉	—	2	1	no	—
72	C₂×C₄×D₁₀₉	—	2	1	no	—
73	C₂×C₄×D₁₀₉	—	2	1	no	—
74	C₂×C₄×D₁₀₉	—	2	1	no	—
75	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
76	D₈₇₂	—	2	2	no	—
77	D₈₇₂	—	2	2	no	—
78	#13182014	—	2	2	no	—
79	#13182014	—	2	2	no	—
80	C₂×#11724784	C₂³	1	1	yes	characteristic
81	C₂×#11724784	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
83	#13182008	D₄	1	1	yes	characteristic
84	#13182008	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₁₀₉	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	#13182006	—	2	1	no	—
92	#13182006	—	2	1	no	—
93	#13182006	D₄	1	1	yes	characteristic
94	#13182006	D₄	1	1	yes	characteristic
95	C₄×Dic₁₀₉	—	2	1	no	—
96	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₀₉	—	2	1	no	—
99	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₁₀₉	—	2	1	no	—
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	—	2	4	no	—
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	C₂²×D₁₀₉	—	8	1	no	—
114	C₂²×D₁₀₉	—	8	1	no	—
115	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	#41	1	2	yes	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	2	1	no	—
121	C₂×C₄×C₁₀₉	—	8	1	no	—
122	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₁₀₉	—	2	4	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	8	1	no	—
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	D₄₃₆	—	2	1	no	—
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	2	1	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	8	1	no	—
137	D₄₃₆	—	8	1	no	—
138	D₄₃₆	—	8	1	no	—
139	D₄₃₆	—	8	1	no	—
140	D₄₃₆	#41	1	2	yes	—
141	C₄×D₁₀₉	—	2	1	no	—
142	C₄×D₁₀₉	—	2	1	no	—
143	C₄×D₁₀₉	—	8	1	no	—
144	C₄×D₁₀₉	—	8	1	no	—
145	C₄×D₁₀₉	—	8	1	no	—
146	C₄×D₁₀₉	—	8	1	no	—
147	C₈×C₁₀₉	—	2	1	no	—
148	C₈×C₁₀₉	—	2	1	no	—
149	#11724784	—	2	1	no	—
150	#11724784	—	2	1	no	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	4	1	no	—
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	2	2	no	—
157	C₂×D₁₀₉	—	2	1	no	—
158	C₂×D₁₀₉	—	2	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	C₂×D₁₀₉	—	4	1	no	—
165	C₂×D₁₀₉	—	4	1	no	—
166	C₂×D₁₀₉	—	4	1	no	—
167	C₂×D₁₀₉	—	8	4	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	—	4	1	no	—
173	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
174	C₄×C₁₀₉	C₂×#41	1	1	yes	characteristic
175	Dic₁₀₉	—	2	1	no	—
176	Dic₁₀₉	—	2	1	no	—
177	Dic₁₀₉	—	2	1	no	—
178	Dic₁₀₉	—	4	1	no	—
179	C₂×C₁₀₉	—	4	2	no	—
180	C₂×C₁₀₉	#525	1	1	yes	characteristic
181	C₂×C₁₀₉	#576	1	1	yes	characteristic
182	C₂×C₁₀₉	#576	1	1	yes	characteristic
183	D₁₀₉	—	4	1	no	—
184	D₁₀₉	—	4	2	no	—
185	D₁₀₉	—	8	1	no	—
186	D₁₀₉	—	8	1	no	—
187	#3122	—	109	1	no	maximal, Sylow
188	C₁₀₉	#3122	1	1	yes	characteristic, complemented, Sylow
189	#540	—	109	1	no	—
190	#518	—	109	1	no	—
191	#482	—	109	1	no	—
192	#482	—	109	1	no	—
193	#478	—	109	1	no	—
194	#478	D₁₀₉	1	1	yes	characteristic, complemented
195	#459	—	109	1	no	—
196	#459	—	109	1	no	—
197	#447	—	109	1	no	—
198	#447	—	109	1	no	—
199	#442	—	109	1	no	—
200	#442	—	109	1	no	—
201	C₄×D₈	—	109	1	no	—
202	C₄×D₈	—	109	1	no	—
203	#433	—	109	1	no	—
204	C₂²×D₄	—	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	#125	C₂×D₁₀₉	1	1	yes	characteristic, complemented
211	#123	—	218	1	no	—
212	#123	—	218	1	no	—
213	#121	—	109	1	no	—
214	#121	—	109	1	no	—
215	#121	—	218	1	no	—
216	#121	—	218	1	no	—
217	C₄×D₄	—	109	1	no	—
218	C₄×D₄	—	109	1	no	—
219	C₄×D₄	—	109	1	no	—
220	C₄×D₄	—	109	1	no	—
221	C₄×D₄	—	109	1	no	—
222	C₄×D₄	—	218	1	no	—
223	C₄×D₄	—	218	1	no	—
224	C₄×D₄	C₂×D₁₀₉	1	1	yes	characteristic
225	#117	—	109	1	no	—
226	#117	—	109	1	no	—
227	C₂×C₂²⋊C₄	—	218	1	no	—
228	C₂×C₂²⋊C₄	—	218	1	no	—
229	#107	—	109	1	no	—
230	#107	C₂×D₁₀₉	1	1	yes	characteristic, complemented
231	#106	—	109	1	no	—
232	#106	C₂×D₁₀₉	1	1	yes	characteristic, complemented
233	C₄⋊C₈	—	109	1	no	—
234	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic, complemented
235	#102	—	109	1	no	—
236	#102	—	109	1	no	—
237	#102	—	109	1	no	—
238	#102	—	109	1	no	—
239	#102	—	109	1	no	—
240	#102	—	109	1	no	—
241	#102	C₂×D₁₀₉	1	1	yes	characteristic
242	#102	C₂×D₁₀₉	1	1	yes	characteristic
243	C₂²⋊C₈	—	109	1	no	—
244	C₂²⋊C₈	—	109	1	no	—
245	#97	—	109	1	no	—
246	C₄×C₈	—	109	1	no	—
247	C₂⁴	—	218	1	no	—
248	C₂×D₄	—	109	1	no	—
249	C₂×D₄	—	109	1	no	—
250	C₂×D₄	—	109	1	no	—
251	C₂×D₄	—	218	1	no	—
252	C₂×D₄	—	218	1	no	—
253	C₂×D₄	—	218	1	no	—
254	C₂×D₄	—	218	2	no	—
255	C₂×D₄	—	218	2	no	—
256	C₂×D₄	—	218	1	no	—
257	C₂×D₄	—	218	1	no	—
258	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	109	1	no	—
261	C₂²×C₄	—	218	1	no	—
262	C₂²×C₄	—	218	1	no	—
263	C₂²×C₄	—	218	1	no	—
264	C₂²×C₄	—	218	1	no	—
265	C₂²×C₄	—	218	1	no	—
266	D₈	—	218	2	no	—
267	D₈	—	218	2	no	—
268	D₈	—	218	2	no	—
269	D₈	—	218	2	no	—
270	C₂×C₈	—	109	1	no	—
271	C₂×C₈	—	109	1	no	—
272	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	109	1	no	—
277	C₄⋊C₄	—	109	1	no	—
278	C₄⋊C₄	—	109	1	no	—
279	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
280	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
281	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	109	1	no	—
284	C₂²⋊C₄	—	109	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	C₂²⋊C₄	—	218	1	no	—
291	C₂²⋊C₄	—	218	1	no	—
292	C₂²⋊C₄	—	218	1	no	—
293	C₄²	—	109	1	no	—
294	C₄²	—	218	1	no	—
295	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	872	1	no	—
298	C₂³	—	872	1	no	—
299	C₂³	—	109	1	no	—
300	C₂³	—	218	4	no	—
301	C₂³	—	218	1	no	—
302	C₂³	—	218	1	no	—
303	C₂³	—	218	1	no	—
304	D₄	—	2	1	no	—
305	D₄	—	872	1	no	—
306	D₄	—	872	1	no	—
307	D₄	—	872	1	no	—
308	D₄	—	872	1	no	—
309	D₄	—	109	2	no	—
310	D₄	—	218	4	no	—
311	D₄	—	218	1	no	—
312	D₄	—	218	1	no	—
313	D₄	—	218	1	no	—
314	D₄	—	218	1	no	—
315	D₄	—	218	1	no	—
316	D₄	#13182030	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	8	1	no	—
321	C₂×C₄	—	872	1	no	—
322	C₂×C₄	—	872	1	no	—
323	C₂×C₄	—	872	1	no	—
324	C₂×C₄	—	872	1	no	—
325	C₂×C₄	—	872	1	no	—
326	C₂×C₄	—	109	1	no	—
327	C₂×C₄	—	109	1	no	—
328	C₂×C₄	—	109	1	no	—
329	C₂×C₄	—	218	1	no	—
330	C₂×C₄	—	218	1	no	—
331	C₂×C₄	—	218	1	no	—
332	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
333	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	218	1	no	—
338	C₈	—	218	1	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	872	4	no	—
343	C₂²	—	436	1	no	—
344	C₂²	—	436	1	no	—
345	C₂²	—	436	1	no	—
346	C₂²	—	436	1	no	—
347	C₂²	—	436	1	no	—
348	C₂²	—	436	1	no	—
349	C₂²	—	436	1	no	—
350	C₂²	—	436	1	no	—
351	C₂²	—	218	1	no	—
352	C₂²	—	218	2	no	—
353	C₂²	—	218	1	no	—
354	C₂²	—	218	1	no	—
355	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	—	4	1	no	—
359	C₄	—	4	1	no	—
360	C₄	—	4	1	no	—
361	C₄	—	436	1	no	—
362	C₄	—	218	1	no	—
363	C₄	—	218	1	no	—
364	C₄	—	218	1	no	—
365	C₄	#41×D₁₀₉	1	1	yes	characteristic
366	C₄	C₂×#13182030	1	1	yes	characteristic
367	C₂	—	4	2	no	—
368	C₂	—	872	1	no	—
369	C₂	—	872	1	no	—
370	C₂	—	436	1	no	—
371	C₂	—	436	2	no	—
372	C₂	#20295539	1	1	yes	characteristic
373	C₂	#20295828	1	1	yes	characteristic
374	C₂	#20295830	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#478×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
hypercenter	#478	D₁₀₉
hypocenter	C₁₀₉	#3122
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #20295632	C₂	semidirect
2. #20295498	C₂	semidirect
3. #20295353	C₂	semidirect
4. #20295352	C₂	semidirect
5. #20295323	C₂	semidirect
6. #20295222	C₂	semidirect
7. #20295221	C₂	semidirect
8. #20295161	C₂	semidirect
9. #20295161	C₂	semidirect
10. #20295137	C₂	semidirect
11. #20295135	C₂	semidirect
12. #20295108	C₂	semidirect
13. #20295106	C₂	semidirect
14. #20295086	C₂	semidirect
15. #478×C₁₀₉	C₂	semidirect
21. #16812696	C₂²	semidirect
31. #16812661	C₂²	semidirect
32. #16812661	C₂²	semidirect
39. #16812632	C₂²	semidirect
40. #16812630	C₂²	semidirect
41. C₄⋊#11724784	C₂²	semidirect
42. #16812621	C₂²	semidirect
43. #16812621	C₂²	semidirect
46. #16812619	C₂²	semidirect
47. #16812619	C₂²	semidirect
50. #16812609	C₂²	semidirect
51. C₈×Dic₁₀₉	C₂²	semidirect
52. #125×C₁₀₉	C₂²	semidirect
54. #107×C₁₀₉	C₂²	semidirect
55. #106×C₁₀₉	C₂²	semidirect
56. C₄⋊C₈×C₁₀₉	C₂²	semidirect
188. C₁₀₉	#3122	semidirect
194. #478	D₁₀₉	semidirect
210. #125	C₂×D₁₀₉	semidirect
230. #107	C₂×D₁₀₉	semidirect
232. #106	C₂×D₁₀₉	semidirect
234. C₄⋊C₈	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295632	2
#20295498	2
#20295353	2
#20295352	2
#20295323	2
#20295222	2
#20295221	2
#20295161	2
#20295137	2
#20295135	2
#20295108	2
#20295106	2
#20295086	2
#478×C₁₀₉	2
#3122	109