Subgroups — #28379804

Order 13952.

456 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₁₀₉⋊#131	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16812786	C₂	1	1	yes	maximal, characteristic, complemented
3	#20296073	C₂	1	2	yes	maximal, complemented
4	#20296071	C₂	1	2	yes	maximal, complemented
5	#20296069	C₂	1	2	yes	maximal, complemented
6	#20296068	C₂	1	2	yes	maximal, complemented
7	#20296041	C₂	1	1	yes	maximal, characteristic, complemented
8	#20296024	C₂	1	1	yes	maximal, characteristic, complemented
9	#20296023	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295992	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295822	C₂	1	1	yes	maximal, characteristic
12	#16812787	C₂²	1	1	yes	characteristic, complemented
13	#16812787	C₂²	1	1	yes	characteristic, complemented
14	#16812786	—	2	1	no	—
15	#16812786	—	2	1	no	—
16	#16812786	—	2	1	no	—
17	#16812786	—	2	1	no	—
18	#16812786	C₂²	1	1	yes	characteristic, complemented
19	#16812786	C₂²	1	2	yes	complemented
20	#16812786	C₂²	1	2	yes	complemented
21	#16812786	C₂²	1	1	yes	characteristic, complemented
22	C₂×#13182039	C₂²	1	1	yes	characteristic
23	D₁₀₉⋊M₁₆	—	2	1	no	—
24	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
25	C₂×#13182038	—	2	1	no	—
26	C₂×D₁₀₉⋊C₈	—	2	1	no	—
27	C₂×D₁₀₉⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²×#11724786	—	2	1	no	—
29	#16812778	C₂²	1	1	yes	characteristic
30	#16812770	C₂²	1	2	yes	complemented
31	#16812769	C₂²	1	2	yes	complemented
32	#16812764	C₂²	1	2	yes	complemented
33	#16812763	C₂²	1	2	yes	complemented
34	#16812763	C₂²	1	2	yes	complemented
35	#16812763	C₂²	1	2	yes	complemented
36	#16812753	C₂²	1	2	yes	complemented
37	C₈×#89137	C₂²	1	2	yes	complemented
38	C₂×#13182028	C₂²	1	1	yes	characteristic
39	#41×D₁₀₉	C₂²	1	1	yes	characteristic
40	#16812732	C₄	1	1	yes	characteristic, complemented
41	#16812729	C₄	1	1	yes	characteristic, complemented
42	#16812728	C₄	1	2	yes	complemented
43	#16812725	C₄	1	1	yes	characteristic, complemented
44	#16812723	C₄	1	2	yes	complemented
45	#16812720	C₂²	1	2	yes	—
46	#16812718	C₂²	1	2	yes	—
47	M₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
48	AGL(1,ℤ/8)×C₁₀₉	C₄	1	1	yes	characteristic, complemented
49	#13182039	—	2	1	no	—
50	#13182039	—	2	1	no	—
51	#13182039	—	2	1	no	—
52	#13182039	C₂³	1	1	yes	characteristic
53	#13182039	C₂³	1	1	yes	characteristic
54	#13182039	C₂³	1	2	yes	—
55	#13182038	—	2	1	no	—
56	#13182038	—	2	2	no	—
57	#13182038	—	2	2	no	—
58	#13182038	—	2	1	no	—
59	#13182038	—	2	1	no	—
60	D₁₀₉⋊C₈	—	2	1	no	—
61	D₁₀₉⋊C₈	—	2	1	no	—
62	D₁₀₉⋊C₈	—	2	1	no	—
63	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
64	D₁₀₉⋊C₈	D₄	1	1	yes	characteristic
65	D₁₀₉⋊C₈	C₂³	1	1	yes	characteristic
66	D₁₀₉⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×#11724786	—	2	1	no	—
68	C₂×#11724786	—	2	2	no	—
69	C₂×#11724786	—	2	2	no	—
70	C₂×#11724786	—	2	1	no	—
71	C₂×#11724786	—	2	1	no	—
72	C₂×#11724786	D₄	1	1	yes	characteristic
73	C₂×#11724786	D₄	1	1	yes	characteristic
74	C₄⋊#89137	—	2	1	no	—
75	C₂²⋊#89137	—	2	1	no	—
76	C₄×#89137	C₂³	1	2	yes	—
77	#13182030	C₂³	1	1	yes	characteristic
78	#13182029	—	2	1	no	—
79	#13182029	C₂×C₄	1	1	yes	characteristic
80	#13182028	—	2	1	no	—
81	#13182028	—	2	1	no	—
82	#13182028	—	2	1	no	—
83	#13182028	C₂×C₄	1	2	yes	—
84	#13182028	C₂³	1	1	yes	characteristic
85	#13182028	C₂³	1	2	yes	—
86	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
87	Q₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
88	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
89	D₄×D₁₀₉	—	2	1	no	—
90	D₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
91	C₂²×Dic₁₀₉	—	2	1	no	—
92	C₂×C₄×D₁₀₉	—	2	1	no	—
93	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
94	Dic₄₃₆	C₂×C₄	1	2	yes	—
95	#13182019	C₂×C₄	1	2	yes	—
96	#13182018	C₂×C₄	1	2	yes	—
97	#13182017	C₂×C₄	1	2	yes	—
98	#13182017	C₂×C₄	1	2	yes	—
99	#13182014	C₂×C₄	1	2	yes	—
100	#13182013	C₂×C₄	1	1	yes	characteristic
101	#13182012	C₂×C₄	1	2	yes	—
102	C₈×D₁₀₉	C₂³	1	2	yes	—
103	#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
104	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
105	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	—
106	D₈×C₁₀₉	C₂×C₄	1	2	yes	—
107	M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	#41	1	1	yes	characteristic
111	C₂×#89137	—	2	2	no	—
112	Q₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
113	D₄×C₁₀₉	—	2	1	no	—
114	D₄×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
116	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
119	C₂²⋊D₁₀₉	—	2	1	no	—
120	C₂²⋊D₁₀₉	—	2	2	no	—
121	C₂²⋊D₁₀₉	—	2	1	no	—
122	C₂²⋊D₁₀₉	—	2	2	no	—
123	C₂²⋊D₁₀₉	—	2	1	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	2	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	2	2	no	—
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	D₄₃₆	—	2	1	no	—
130	D₄₃₆	C₂²×C₄	1	1	yes	characteristic
131	C₄×D₁₀₉	—	2	1	no	—
132	C₄×D₁₀₉	—	2	1	no	—
133	C₄×D₁₀₉	—	2	1	no	—
134	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
136	C₄×D₁₀₉	#41	1	1	yes	characteristic
137	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
138	Dic₂₁₈	—	2	1	no	—
139	Dic₂₁₈	—	2	1	no	—
140	Dic₂₁₈	C₂²×C₄	1	1	yes	characteristic
141	Dic₂₁₈	C₂²×C₄	1	2	yes	—
142	#11724786	—	2	2	no	—
143	#11724786	—	2	1	no	—
144	#11724786	—	2	1	no	—
145	#11724786	C₂×D₄	1	1	yes	characteristic
146	#11724786	C₂×D₄	1	1	yes	characteristic
147	#11724786	C₂×D₄	1	1	yes	characteristic
148	#11724786	C₂×D₄	1	1	yes	characteristic
149	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
150	#11724784	C₂²×C₄	1	2	yes	—
151	C₂²×C₁₀₉	—	2	2	no	—
152	C₂²×C₁₀₉	—	2	1	no	—
153	C₂²×C₁₀₉	—	4	2	no	—
154	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
155	C₂×D₁₀₉	—	2	1	no	—
156	C₂×D₁₀₉	—	4	1	no	—
157	C₂×D₁₀₉	—	4	2	no	—
158	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
159	C₂×D₁₀₉	C₂×#41	1	1	yes	characteristic
160	#89137	—	4	2	no	—
161	C₄×C₁₀₉	—	2	1	no	—
162	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
163	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
164	Dic₁₀₉	—	2	1	no	—
165	Dic₁₀₉	—	2	2	no	—
166	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
167	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₁₀₉	—	2	1	no	—
169	C₂×C₁₀₉	—	4	1	no	—
170	C₂×C₁₀₉	—	4	2	no	—
171	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
172	D₁₀₉	—	2	1	no	—
173	D₁₀₉	—	2	1	no	—
174	D₁₀₉	—	4	1	no	—
175	#2955	—	109	1	no	maximal, Sylow
176	C₁₀₉	#2955	1	1	yes	characteristic, complemented, Sylow
177	#575	—	109	1	no	—
178	C₂×#131	—	109	1	no	—
179	C₂×#131	—	109	1	no	—
180	#444	—	109	2	no	—
181	#444	—	109	2	no	—
182	#443	—	109	2	no	—
183	#443	—	109	2	no	—
184	#430	—	109	1	no	—
185	#421	—	109	1	no	—
186	#421	—	109	1	no	—
187	#405	—	109	1	no	—
188	C₂×#41	—	109	1	no	—
189	C₂×#41	—	109	1	no	—
190	#137	—	109	1	no	—
191	#137	—	109	2	no	—
192	#137	—	109	1	no	—
193	AGL(1,ℤ/8)	—	109	1	no	—
194	AGL(1,ℤ/8)	—	109	2	no	—
195	AGL(1,ℤ/8)	#89137	1	1	yes	characteristic, complemented
196	#135	—	109	2	no	—
197	#135	—	109	2	no	—
198	#131	—	109	2	no	—
199	#131	—	109	1	no	—
200	#131	—	109	1	no	—
201	#131	—	109	2	no	—
202	#131	—	109	1	no	—
203	#131	—	109	1	no	—
204	#131	—	218	1	no	—
205	#131	—	218	1	no	—
206	#131	—	218	1	no	—
207	#131	—	218	1	no	—
208	C₂×M₁₆	—	109	1	no	—
209	C₂×M₁₆	—	109	1	no	—
210	C₂×M₁₆	—	109	1	no	—
211	C₂×M₁₆	—	218	1	no	—
212	C₂×M₁₆	—	218	1	no	—
213	C₂²×C₈	—	109	1	no	—
214	C₂²×C₈	—	218	1	no	—
215	C₂²×C₈	—	218	1	no	—
216	#117	—	109	1	no	—
217	#108	—	109	2	no	—
218	#108	—	109	2	no	—
219	#104	—	109	2	no	—
220	#104	—	109	2	no	—
221	#104	—	109	2	no	—
222	#104	—	109	2	no	—
223	#97	—	109	2	no	—
224	C₄×C₈	—	109	2	no	—
225	#41	—	109	1	no	—
226	#41	—	109	2	no	—
227	#41	—	109	2	no	—
228	#41	—	109	1	no	—
229	#41	—	109	1	no	—
230	#41	—	218	1	no	—
231	#41	—	218	1	no	—
232	#41	—	218	1	no	—
233	#41	—	218	1	no	—
234	#41	C₂×#89137	1	1	yes	characteristic
235	C₂×Q₈	—	109	1	no	—
236	C₂×Q₈	—	109	1	no	—
237	C₂×D₄	—	109	1	no	—
238	C₂×D₄	—	218	1	no	—
239	C₂×D₄	—	218	1	no	—
240	C₂×D₄	C₂×#89137	1	1	yes	characteristic
241	C₂²×C₄	—	109	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	C₂²×C₄	—	218	1	no	—
244	Q₁₆	—	109	2	no	—
245	Q₁₆	—	109	2	no	—
246	SD₁₆	—	109	2	no	—
247	SD₁₆	—	109	2	no	—
248	SD₁₆	—	109	2	no	—
249	SD₁₆	C₂×#89137	1	2	yes	—
250	D₈	—	109	2	no	—
251	D₈	C₂×#89137	1	2	yes	—
252	M₁₆	—	109	1	no	—
253	M₁₆	—	109	2	no	—
254	M₁₆	—	109	1	no	—
255	M₁₆	—	109	1	no	—
256	M₁₆	—	109	2	no	—
257	M₁₆	—	218	2	no	—
258	M₁₆	—	218	1	no	—
259	M₁₆	—	218	2	no	—
260	M₁₆	—	218	1	no	—
261	M₁₆	—	218	1	no	—
262	M₁₆	—	218	1	no	—
263	M₁₆	—	218	1	no	—
264	M₁₆	—	218	1	no	—
265	M₁₆	C₂×#89137	1	1	yes	characteristic
266	C₂×C₈	—	109	1	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	C₂×C₈	—	109	1	no	—
272	C₂×C₈	—	109	2	no	—
273	C₂×C₈	—	218	2	no	—
274	C₂×C₈	—	218	1	no	—
275	C₂×C₈	—	218	2	no	—
276	C₂×C₈	—	218	1	no	—
277	C₂×C₈	—	218	1	no	—
278	C₂×C₈	—	218	1	no	—
279	C₂×C₈	—	218	1	no	—
280	C₂×C₈	—	218	1	no	—
281	C₄⋊C₄	—	218	1	no	—
282	C₂²⋊C₄	—	218	1	no	—
283	C₄²	—	109	2	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	109	1	no	—
286	C₂³	—	218	1	no	—
287	Q₈	—	109	1	no	—
288	Q₈	—	109	2	no	—
289	Q₈	—	218	1	no	—
290	Q₈	—	218	1	no	—
291	Q₈	C₂²×#89137	1	1	yes	characteristic
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	109	1	no	—
295	D₄	—	218	1	no	—
296	D₄	—	218	1	no	—
297	D₄	—	218	1	no	—
298	D₄	—	218	2	no	—
299	D₄	—	218	2	no	—
300	D₄	—	218	1	no	—
301	D₄	C₂²×#89137	1	2	yes	—
302	D₄	C₂²×#89137	1	1	yes	characteristic
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	109	1	no	—
305	C₂×C₄	—	109	1	no	—
306	C₂×C₄	—	109	1	no	—
307	C₂×C₄	—	109	1	no	—
308	C₂×C₄	—	109	1	no	—
309	C₂×C₄	—	218	1	no	—
310	C₂×C₄	—	218	1	no	—
311	C₂×C₄	—	218	1	no	—
312	C₂×C₄	—	218	2	no	—
313	C₂×C₄	—	218	2	no	—
314	C₂×C₄	—	218	2	no	—
315	C₂×C₄	—	218	1	no	—
316	C₂×C₄	—	218	1	no	—
317	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
318	C₈	—	109	1	no	—
319	C₈	—	109	1	no	—
320	C₈	—	109	1	no	—
321	C₈	—	109	1	no	—
322	C₈	—	109	2	no	—
323	C₈	—	218	2	no	—
324	C₈	—	218	1	no	—
325	C₈	—	218	1	no	—
326	C₈	C₂²×#89137	1	2	yes	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	4	2	no	—
330	C₂²	—	109	1	no	—
331	C₂²	—	109	1	no	—
332	C₂²	—	436	1	no	—
333	C₂²	—	436	2	no	—
334	C₂²	—	218	1	no	—
335	C₂²	D₄×#89137	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	109	1	no	—
338	C₄	—	109	1	no	—
339	C₄	—	436	2	no	—
340	C₄	—	218	1	no	—
341	C₄	—	218	2	no	—
342	C₄	D₄×#89137	1	1	yes	characteristic
343	C₄	C₂³×#89137	1	1	yes	characteristic
344	C₂	—	2	1	no	—
345	C₂	—	4	2	no	—
346	C₂	—	4	1	no	—
347	C₂	—	436	1	no	—
348	C₂	—	218	1	no	—
349	C₂	—	218	1	no	—
350	C₂	C₂×D₄×#89137	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#89137
commutator	C₄×C₁₀₉	C₂³×C₄
Frattini	C₄	C₂³×#89137
Fitting	AGL(1,ℤ/8)×C₁₀₉	C₄
socle	C₂×C₁₀₉	C₂×C₄×D₄
prime radical	#20295822	C₂
cosocle	C₄×D₁₀₉	C₂⁴
hypercenter	AGL(1,ℤ/8)	#89137
hypocenter	C₁₀₉	#2955
LCS	G ▷ C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. D₁₀₉⋊#131	C₂	semidirect
2. C₂×#16812786	C₂	semidirect
3. #20296073	C₂	semidirect
4. #20296071	C₂	semidirect
5. #20296069	C₂	semidirect
6. #20296068	C₂	semidirect
7. #20296041	C₂	semidirect
8. #20296024	C₂	semidirect
9. #20296023	C₂	semidirect
10. #20295992	C₂	semidirect
12. #16812787	C₂²	semidirect
13. #16812787	C₂²	semidirect
18. #16812786	C₂²	semidirect
19. #16812786	C₂²	semidirect
20. #16812786	C₂²	semidirect
21. #16812786	C₂²	semidirect
30. #16812770	C₂²	semidirect
31. #16812769	C₂²	semidirect
32. #16812764	C₂²	semidirect
33. #16812763	C₂²	semidirect
34. #16812763	C₂²	semidirect
35. #16812763	C₂²	semidirect
36. #16812753	C₂²	semidirect
37. C₈×#89137	C₂²	semidirect
40. #16812732	C₄	semidirect
41. #16812729	C₄	semidirect
42. #16812728	C₄	semidirect
43. #16812725	C₄	semidirect
44. #16812723	C₄	semidirect
48. AGL(1,ℤ/8)×C₁₀₉	C₄	semidirect
176. C₁₀₉	#2955	semidirect
195. AGL(1,ℤ/8)	#89137	semidirect

Maximal subgroups

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

subgroup	index
D₁₀₉⋊#131	2
C₂×#16812786	2
#20296073	2
#20296071	2
#20296069	2
#20296068	2
#20296041	2
#20296024	2
#20296023	2
#20295992	2
#20295822	2
#2955	109