Subgroups — #28372751

Order 13952.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295826	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295824	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16812719	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16812707	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295300	C₂	1	2	yes	maximal, complemented
6	#20295295	C₂	1	2	yes	maximal, complemented
7	#20295292	C₂	1	2	yes	maximal, complemented
8	#20295288	C₂	1	2	yes	maximal, complemented
9	C₂×#16812634	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295008	C₂	1	1	yes	maximal, characteristic, complemented
11	#413×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13182029	C₂²	1	1	yes	characteristic
13	C₂×#13182029	C₂²	1	1	yes	characteristic
14	#16812732	—	2	1	no	—
15	#16812732	C₂²	1	2	yes	—
16	#16812730	C₂²	1	2	yes	—
17	#16812729	—	2	1	no	—
18	#16812727	—	2	1	no	—
19	#16812727	C₂²	1	2	yes	—
20	#16812725	C₂²	1	2	yes	—
21	#16812724	—	2	1	no	—
22	#16812721	—	2	2	no	—
23	#16812719	—	2	2	no	—
24	#16812719	—	2	1	no	—
25	#16812719	—	2	1	no	—
26	#16812719	—	2	1	no	—
27	#16812719	—	2	1	no	—
28	C₂×#13182019	C₂²	1	1	yes	characteristic, complemented
29	C₂×#13182017	C₂²	1	1	yes	characteristic, complemented
30	C₂×#13182016	C₂²	1	1	yes	characteristic, complemented
31	C₂×#13182014	C₂²	1	1	yes	characteristic, complemented
32	#16812707	—	2	1	no	—
33	#16812707	—	2	1	no	—
34	#16812707	C₂²	1	2	yes	—
35	#16812707	C₂²	1	2	yes	—
36	C₂×#13182013	C₂²	1	1	yes	characteristic
37	C₂×#13182012	—	2	1	no	—
38	C₂²×#11724784	C₂²	1	1	yes	characteristic
39	C₂×C₈×D₁₀₉	—	2	1	no	—
40	#16812634	C₂²	1	2	yes	complemented
41	#16812634	C₂²	1	2	yes	complemented
42	#16812617	C₂²	1	2	yes	complemented
43	#16812615	C₂²	1	2	yes	complemented
44	C₂×#41×C₁₀₉	C₂²	1	1	yes	characteristic
45	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
46	C₂×M₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
47	#101×C₁₀₉	C₂²	1	2	yes	complemented
48	#100×C₁₀₉	C₂²	1	2	yes	complemented
49	#13182029	—	2	1	no	—
50	#13182029	—	2	1	no	—
51	#13182029	—	2	1	no	—
52	#13182029	—	2	1	no	—
53	#13182029	—	2	1	no	—
54	#13182029	—	2	1	no	—
55	#13182029	D₄	1	2	yes	—
56	#13182029	C₂³	1	2	yes	—
57	C₂×Dic₂₁₈	D₄	1	1	yes	characteristic
58	C₂×Dic₂₁₈	C₂³	1	1	yes	characteristic
59	C₂×D₄₃₆	D₄	1	1	yes	characteristic
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	—	2	1	no	—
65	Dic₄₃₆	—	2	2	no	—
66	#13182019	—	2	2	no	—
67	#13182019	—	2	1	no	—
68	#13182019	—	2	1	no	—
69	#13182018	—	2	2	no	—
70	#13182018	—	2	2	no	—
71	#13182017	—	2	2	no	—
72	#13182017	—	2	1	no	—
73	#13182017	—	2	1	no	—
74	#13182016	—	2	2	no	—
75	#13182016	—	2	1	no	—
76	#13182016	—	2	1	no	—
77	D₈₇₂	—	2	2	no	—
78	#13182014	—	2	2	no	—
79	#13182014	—	2	1	no	—
80	#13182014	—	2	1	no	—
81	#13182013	—	2	1	no	—
82	#13182013	C₂³	1	2	yes	—
83	#13182012	—	2	2	no	—
84	#13182012	—	2	2	no	—
85	C₂×#11724784	—	2	1	no	—
86	C₂×#11724784	D₄	1	1	yes	characteristic
87	C₂×#11724784	D₄	1	1	yes	characteristic
88	C₂×#11724784	D₄	1	1	yes	characteristic
89	C₂×#11724784	D₄	1	1	yes	characteristic
90	C₂×#11724784	C₂³	1	1	yes	characteristic
91	C₂×#11724784	C₂³	1	1	yes	characteristic
92	C₈×D₁₀₉	—	2	2	no	—
93	C₈×D₁₀₉	—	2	2	no	—
94	#41×C₁₀₉	—	2	1	no	—
95	#41×C₁₀₉	—	2	1	no	—
96	#41×C₁₀₉	—	2	1	no	—
97	#41×C₁₀₉	—	2	1	no	—
98	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
99	C₂×D₄×C₁₀₉	—	2	1	no	—
100	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂²×C₄×C₁₀₉	—	2	1	no	—
102	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
103	M₁₆×C₁₀₉	—	2	1	no	—
104	M₁₆×C₁₀₉	—	2	1	no	—
105	M₁₆×C₁₀₉	C₂³	1	2	yes	—
106	M₁₆×C₁₀₉	C₂³	1	2	yes	—
107	C₂×C₈×C₁₀₉	—	2	1	no	—
108	C₂×C₈×C₁₀₉	—	2	1	no	—
109	C₂³×C₁₀₉	—	2	1	no	—
110	C₂³×C₁₀₉	#41	1	1	yes	characteristic
111	C₂²×D₁₀₉	—	2	1	no	—
112	C₂²×D₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	D₄×C₁₀₉	—	4	1	no	—
118	D₄×C₁₀₉	—	4	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	—	4	1	no	—
121	C₂×C₄×C₁₀₉	—	4	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	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₁₀₉	#41	1	1	yes	characteristic
127	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₁₀₉	—	2	2	no	—
129	C₂²⋊D₁₀₉	—	2	2	no	—
130	C₂²⋊D₁₀₉	—	4	1	no	—
131	C₂²⋊D₁₀₉	—	4	1	no	—
132	C₂×Dic₁₀₉	—	2	1	no	—
133	C₂×Dic₁₀₉	—	2	1	no	—
134	D₄₃₆	—	2	1	no	—
135	D₄₃₆	—	2	1	no	—
136	D₄₃₆	—	2	1	no	—
137	D₄₃₆	C₂×D₄	1	2	yes	—
138	C₄×D₁₀₉	—	2	2	no	—
139	C₄×D₁₀₉	—	2	2	no	—
140	C₄×D₁₀₉	—	4	1	no	—
141	C₄×D₁₀₉	—	4	1	no	—
142	Dic₂₁₈	—	2	1	no	—
143	Dic₂₁₈	—	2	1	no	—
144	Dic₂₁₈	—	2	1	no	—
145	Dic₂₁₈	C₂×D₄	1	2	yes	—
146	C₈×C₁₀₉	—	2	2	no	—
147	C₈×C₁₀₉	—	2	2	no	—
148	#11724784	—	2	2	no	—
149	#11724784	C₂×D₄	1	1	yes	characteristic
150	#11724784	C₂×D₄	1	1	yes	characteristic
151	#11724784	C₂×D₄	1	1	yes	characteristic
152	#11724784	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₁₀₉	—	4	1	no	—
154	C₂²×C₁₀₉	—	4	1	no	—
155	C₂²×C₁₀₉	—	8	1	no	—
156	C₂²×C₁₀₉	#121	1	1	yes	characteristic
157	C₂²×C₁₀₉	#121	1	1	yes	characteristic
158	C₂²×C₁₀₉	C₂×#41	1	1	yes	characteristic
159	C₂×D₁₀₉	—	2	2	no	—
160	C₂×D₁₀₉	—	4	2	no	—
161	C₂×D₁₀₉	—	4	1	no	—
162	C₂×D₁₀₉	—	8	1	no	—
163	C₄×C₁₀₉	—	4	1	no	—
164	C₄×C₁₀₉	#121	1	1	yes	characteristic
165	C₄×C₁₀₉	#121	1	1	yes	characteristic
166	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
168	Dic₁₀₉	—	2	2	no	—
169	Dic₁₀₉	—	4	1	no	—
170	C₂×C₁₀₉	—	2	1	no	—
171	C₂×C₁₀₉	—	2	1	no	—
172	C₂×C₁₀₉	—	2	1	no	—
173	C₂×C₁₀₉	—	8	1	no	—
174	C₂×C₁₀₉	C₂×#121	1	1	yes	characteristic
175	D₁₀₉	—	4	2	no	—
176	D₁₀₉	—	8	1	no	—
177	#3067	—	109	1	no	maximal, Sylow
178	C₁₀₉	#3067	1	1	yes	characteristic, complemented, Sylow
179	#575	—	109	1	no	—
180	#575	—	109	1	no	—
181	C₂×#135	—	109	1	no	—
182	C₂×#131	—	109	1	no	—
183	#473	—	109	2	no	—
184	#472	—	109	2	no	—
185	#471	—	109	2	no	—
186	#471	—	109	2	no	—
187	C₂×#108	—	109	1	no	—
188	#413	—	109	1	no	—
189	#413	D₁₀₉	1	1	yes	characteristic, complemented
190	C₂×#41	—	109	1	no	—
191	C₂×#41	—	109	1	no	—
192	C₂×#41	C₂×D₁₀₉	1	1	yes	characteristic
193	#137	—	109	2	no	—
194	#137	—	109	2	no	—
195	#137	—	218	1	no	—
196	#137	—	218	1	no	—
197	AGL(1,ℤ/8)	—	109	2	no	—
198	AGL(1,ℤ/8)	—	109	2	no	—
199	AGL(1,ℤ/8)	—	218	1	no	—
200	AGL(1,ℤ/8)	—	218	1	no	—
201	#135	—	218	2	no	—
202	#135	—	218	1	no	—
203	#135	—	218	1	no	—
204	#135	—	218	1	no	—
205	#135	—	218	1	no	—
206	#135	—	218	2	no	—
207	C₂×Q₁₆	—	109	1	no	—
208	C₂×SD₁₆	—	109	1	no	—
209	C₂×SD₁₆	—	109	1	no	—
210	C₂×D₈	—	109	1	no	—
211	#131	—	109	2	no	—
212	#131	—	109	2	no	—
213	#131	—	218	1	no	—
214	#131	—	218	1	no	—
215	C₂×M₁₆	—	109	1	no	—
216	C₂×M₁₆	—	218	1	no	—
217	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
218	C₂×M₁₆	C₂×D₁₀₉	1	1	yes	characteristic
219	C₂²×C₈	—	109	1	no	—
220	C₂²×C₈	—	218	1	no	—
221	#108	—	109	2	no	—
222	#108	—	109	2	no	—
223	#101	—	109	2	no	—
224	#101	C₂×D₁₀₉	1	2	yes	complemented
225	#100	—	109	2	no	—
226	#100	C₂×D₁₀₉	1	2	yes	complemented
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	—	109	2	no	—
232	#41	—	109	2	no	—
233	#41	—	218	1	no	—
234	#41	—	218	1	no	—
235	#41	—	218	1	no	—
236	#41	—	218	1	no	—
237	#41	—	218	1	no	—
238	#41	—	218	1	no	—
239	C₂×Q₈	—	109	1	no	—
240	C₂×Q₈	—	109	1	no	—
241	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	109	1	no	—
244	C₂×D₄	—	109	1	no	—
245	C₂×D₄	—	218	1	no	—
246	C₂×D₄	—	218	1	no	—
247	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	218	1	no	—
250	C₂²×C₄	—	218	1	no	—
251	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
252	Q₁₆	—	218	2	no	—
253	Q₁₆	—	218	1	no	—
254	Q₁₆	—	218	1	no	—
255	Q₁₆	—	218	2	no	—
256	SD₁₆	—	218	2	no	—
257	SD₁₆	—	218	2	no	—
258	SD₁₆	—	218	1	no	—
259	SD₁₆	—	218	1	no	—
260	SD₁₆	—	218	1	no	—
261	SD₁₆	—	218	1	no	—
262	SD₁₆	—	218	2	no	—
263	SD₁₆	—	218	2	no	—
264	D₈	—	218	2	no	—
265	D₈	—	218	1	no	—
266	D₈	—	218	1	no	—
267	D₈	—	218	2	no	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	109	2	no	—
271	M₁₆	—	218	2	no	—
272	M₁₆	—	218	2	no	—
273	M₁₆	—	218	1	no	—
274	M₁₆	C₂²×D₁₀₉	1	2	yes	—
275	M₁₆	C₂²×D₁₀₉	1	2	yes	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	109	1	no	—
279	C₂×C₈	—	109	1	no	—
280	C₂×C₈	—	109	1	no	—
281	C₂×C₈	—	109	1	no	—
282	C₂×C₈	—	109	1	no	—
283	C₂×C₈	—	109	1	no	—
284	C₂×C₈	—	218	2	no	—
285	C₂×C₈	—	218	2	no	—
286	C₂×C₈	—	218	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	218	1	no	—
289	C₂³	—	218	1	no	—
290	C₂³	#13182029	1	1	yes	characteristic
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	109	2	no	—
294	Q₈	—	218	1	no	—
295	Q₈	—	218	1	no	—
296	Q₈	—	218	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	109	2	no	—
302	D₄	—	436	1	no	—
303	D₄	—	436	1	no	—
304	D₄	—	218	2	no	—
305	D₄	—	218	2	no	—
306	D₄	—	218	1	no	—
307	D₄	—	218	1	no	—
308	D₄	—	218	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	436	1	no	—
313	C₂×C₄	—	436	1	no	—
314	C₂×C₄	—	218	1	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	2	no	—
317	C₂×C₄	—	218	2	no	—
318	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
319	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
321	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
322	C₂×C₄	#13182029	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	109	1	no	—
327	C₈	—	109	1	no	—
328	C₈	—	109	1	no	—
329	C₈	—	109	1	no	—
330	C₈	—	218	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	872	1	no	—
335	C₂²	—	436	2	no	—
336	C₂²	—	436	1	no	—
337	C₂²	—	218	2	no	—
338	C₂²	#16812672	1	1	yes	characteristic
339	C₂²	#16812672	1	1	yes	characteristic
340	C₂²	C₂×#13182029	1	1	yes	characteristic
341	C₄	—	4	1	no	—
342	C₄	—	436	1	no	—
343	C₄	—	218	2	no	—
344	C₄	#16812672	1	1	yes	characteristic
345	C₄	#16812672	1	1	yes	characteristic
346	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
347	C₄	C₂×D₄×D₁₀₉	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	872	1	no	—
353	C₂	—	436	2	no	—
354	C₂	C₂×#16812672	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16812672
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#413×C₁₀₉	C₂
socle	C₂×C₁₀₉	C₂×#121
hypercenter	#413	D₁₀₉
hypocenter	C₁₀₉	#3067
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

Complements

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

N	H ≅ G/N	type
1. #20295826	C₂	semidirect
2. #20295824	C₂	semidirect
3. C₂×#16812719	C₂	semidirect
4. C₂×#16812707	C₂	semidirect
5. #20295300	C₂	semidirect
6. #20295295	C₂	semidirect
7. #20295292	C₂	semidirect
8. #20295288	C₂	semidirect
9. C₂×#16812634	C₂	semidirect
10. #20295008	C₂	semidirect
11. #413×C₁₀₉	C₂	semidirect
28. C₂×#13182019	C₂²	semidirect
29. C₂×#13182017	C₂²	semidirect
30. C₂×#13182016	C₂²	semidirect
31. C₂×#13182014	C₂²	semidirect
40. #16812634	C₂²	semidirect
41. #16812634	C₂²	semidirect
42. #16812617	C₂²	semidirect
43. #16812615	C₂²	semidirect
47. #101×C₁₀₉	C₂²	semidirect
48. #100×C₁₀₉	C₂²	semidirect
178. C₁₀₉	#3067	semidirect
189. #413	D₁₀₉	semidirect
224. #101	C₂×D₁₀₉	semidirect
226. #100	C₂×D₁₀₉	semidirect

Maximal subgroups

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

subgroup	index
#20295826	2
#20295824	2
C₂×#16812719	2
C₂×#16812707	2
#20295300	2
#20295295	2
#20295292	2
#20295288	2
C₂×#16812634	2
#20295008	2
#413×C₁₀₉	2
#3067	109