Subgroups — #28379806

Order 13952.

458 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20296105	C₂	1	1	yes	maximal, characteristic, complemented
2	#20296104	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	D₁₀₉⋊#104	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#16812763	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×#89137	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295822	C₂	1	1	yes	maximal, characteristic
12	#16812787	C₂²	1	2	yes	complemented
13	#16812786	—	2	1	no	—
14	#16812786	—	2	1	no	—
15	#16812786	—	2	2	no	—
16	#16812786	C₂²	1	2	yes	complemented
17	#16812786	C₂²	1	2	yes	complemented
18	#16812786	C₂²	1	2	yes	complemented
19	C₂×#13182039	—	2	1	no	—
20	C₂×#13182039	C₂²	1	1	yes	characteristic
21	D₁₀₉⋊M₁₆	—	2	1	no	—
22	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
23	D₁₀₉⋊M₁₆	C₂²	1	1	yes	characteristic
24	C₂×#13182038	—	2	2	no	—
25	C₂×C₄×#89137	C₂²	1	1	yes	characteristic
26	#16812770	C₂²	1	2	yes	complemented
27	#16812769	C₂²	1	2	yes	complemented
28	#16812764	C₂²	1	1	yes	characteristic, complemented
29	#16812764	C₂²	1	1	yes	characteristic, complemented
30	#16812763	C₂²	1	2	yes	complemented
31	#16812763	C₂²	1	2	yes	complemented
32	#16812763	C₂²	1	1	yes	characteristic, complemented
33	#16812763	C₂²	1	1	yes	characteristic, complemented
34	#16812753	C₂²	1	2	yes	complemented
35	C₈×#89137	C₂²	1	2	yes	complemented
36	C₂×#13182028	C₂²	1	1	yes	characteristic
37	#41×D₁₀₉	C₂²	1	1	yes	characteristic
38	#16812732	C₄	1	1	yes	characteristic, complemented
39	#16812729	C₄	1	1	yes	characteristic, complemented
40	#16812728	C₄	1	2	yes	complemented
41	#16812725	C₄	1	1	yes	characteristic, complemented
42	#16812723	C₄	1	2	yes	complemented
43	#16812720	C₂²	1	2	yes	—
44	#16812718	C₂²	1	2	yes	—
45	M₁₆×D₁₀₉	C₂²	1	1	yes	characteristic
46	AGL(1,ℤ/8)×C₁₀₉	C₄	1	1	yes	characteristic, complemented
47	C₂²×#89137	—	2	1	no	—
48	#13182039	—	2	1	no	—
49	#13182039	—	2	2	no	—
50	#13182039	D₄	1	1	yes	characteristic
51	#13182039	D₄	1	1	yes	characteristic
52	#13182039	C₂³	1	2	yes	—
53	#13182039	C₂³	1	1	yes	characteristic
54	#13182039	C₂³	1	1	yes	characteristic
55	#13182038	—	2	1	no	—
56	#13182038	—	2	2	no	—
57	#13182038	—	2	2	no	—
58	#13182038	—	2	2	no	—
59	#13182038	D₄	1	2	yes	—
60	D₁₀₉⋊C₈	—	2	1	no	—
61	D₁₀₉⋊C₈	—	2	2	no	—
62	D₁₀₉⋊C₈	C₂³	1	2	yes	—
63	C₂×#11724786	—	2	1	no	—
64	C₂×#11724786	—	2	2	no	—
65	C₂×#11724786	—	2	2	no	—
66	C₂×#11724786	—	2	2	no	—
67	C₄×#89137	—	2	1	no	—
68	C₄×#89137	C₂³	1	1	yes	characteristic
69	C₄×#89137	C₂³	1	1	yes	characteristic
70	#13182030	C₂³	1	1	yes	characteristic
71	#13182029	—	2	1	no	—
72	#13182029	C₂×C₄	1	1	yes	characteristic
73	#13182028	—	2	1	no	—
74	#13182028	—	2	1	no	—
75	#13182028	—	2	1	no	—
76	#13182028	C₂×C₄	1	2	yes	—
77	#13182028	C₂³	1	1	yes	characteristic
78	#13182028	C₂³	1	2	yes	—
79	C₂×Dic₂₁₈	C₂×C₄	1	1	yes	characteristic
80	Q₈×D₁₀₉	C₂×C₄	1	1	yes	characteristic
81	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
82	D₄×D₁₀₉	—	2	1	no	—
83	D₄×D₁₀₉	C₂×C₄	1	1	yes	characteristic
84	C₂²×Dic₁₀₉	—	2	1	no	—
85	C₂×C₄×D₁₀₉	—	2	1	no	—
86	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
87	Dic₄₃₆	C₂×C₄	1	2	yes	complemented
88	#13182019	C₂×C₄	1	2	yes	complemented
89	#13182018	C₂×C₄	1	2	yes	complemented
90	#13182017	C₂×C₄	1	2	yes	complemented
91	#13182017	C₂×C₄	1	2	yes	complemented
92	#13182014	C₂×C₄	1	2	yes	complemented
93	#13182013	C₂×C₄	1	1	yes	characteristic
94	#13182012	C₂×C₄	1	2	yes	—
95	C₈×D₁₀₉	C₂³	1	2	yes	—
96	#41×C₁₀₉	C₂×C₄	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
98	SD₁₆×C₁₀₉	C₂×C₄	1	2	yes	complemented
99	D₈×C₁₀₉	C₂×C₄	1	2	yes	complemented
100	M₁₆×C₁₀₉	C₂×C₄	1	1	yes	characteristic
101	C₂³×C₁₀₉	—	2	1	no	—
102	C₂²×D₁₀₉	—	2	1	no	—
103	C₂²×D₁₀₉	#41	1	1	yes	characteristic
104	C₂×#89137	—	2	1	no	—
105	C₂×#89137	—	2	1	no	—
106	C₂×#89137	—	4	2	no	—
107	Q₈×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
108	D₄×C₁₀₉	—	2	1	no	—
109	D₄×C₁₀₉	—	2	1	no	—
110	D₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₁₀₉	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₁₀₉	—	2	1	no	—
113	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₁₀₉	—	2	1	no	—
115	C₂²⋊D₁₀₉	—	2	2	no	—
116	C₂²⋊D₁₀₉	—	2	1	no	—
117	C₂²⋊D₁₀₉	—	2	2	no	—
118	C₂²⋊D₁₀₉	—	2	1	no	—
119	C₂×Dic₁₀₉	—	2	1	no	—
120	C₂×Dic₁₀₉	—	2	2	no	—
121	C₂×Dic₁₀₉	—	2	1	no	—
122	C₂×Dic₁₀₉	—	2	2	no	—
123	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
124	D₄₃₆	—	2	1	no	—
125	D₄₃₆	C₂²×C₄	1	1	yes	characteristic
126	C₄×D₁₀₉	—	2	1	no	—
127	C₄×D₁₀₉	—	2	1	no	—
128	C₄×D₁₀₉	—	2	1	no	—
129	C₄×D₁₀₉	C₂²×C₄	1	1	yes	characteristic
130	C₄×D₁₀₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₁₀₉	#41	1	1	yes	characteristic
132	C₄×D₁₀₉	C₂⁴	1	1	yes	characteristic
133	Dic₂₁₈	—	2	1	no	—
134	Dic₂₁₈	—	2	1	no	—
135	Dic₂₁₈	C₂²×C₄	1	1	yes	characteristic
136	Dic₂₁₈	C₂²×C₄	1	2	yes	—
137	#11724786	—	2	1	no	—
138	#11724786	—	2	1	no	—
139	#11724786	—	2	2	no	—
140	#11724786	C₂×D₄	1	2	yes	—
141	#11724786	C₂×D₄	1	2	yes	—
142	C₈×C₁₀₉	C₂²×C₄	1	2	yes	—
143	#11724784	C₂²×C₄	1	2	yes	—
144	C₂²×C₁₀₉	—	2	2	no	—
145	C₂²×C₁₀₉	—	2	1	no	—
146	C₂²×C₁₀₉	—	8	1	no	—
147	C₂²×C₁₀₉	C₄×D₄	1	1	yes	characteristic
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	4	1	no	—
150	C₂×D₁₀₉	—	8	1	no	—
151	C₂×D₁₀₉	C₄×D₄	1	1	yes	characteristic
152	C₂×D₁₀₉	C₂×#41	1	1	yes	characteristic
153	#89137	—	2	2	no	—
154	#89137	—	4	1	no	—
155	C₄×C₁₀₉	—	2	1	no	—
156	C₄×C₁₀₉	C₄×D₄	1	1	yes	characteristic
157	C₄×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
158	Dic₁₀₉	—	2	1	no	—
159	Dic₁₀₉	—	2	2	no	—
160	Dic₁₀₉	C₄×D₄	1	1	yes	characteristic
161	Dic₁₀₉	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₁₀₉	—	2	1	no	—
163	C₂×C₁₀₉	—	4	1	no	—
164	C₂×C₁₀₉	—	4	2	no	—
165	C₂×C₁₀₉	C₂×C₄×D₄	1	1	yes	characteristic
166	D₁₀₉	—	2	1	no	—
167	D₁₀₉	—	2	1	no	—
168	D₁₀₉	—	4	1	no	—
169	#2956	—	109	1	no	maximal, Sylow
170	C₁₀₉	#2956	1	1	yes	characteristic, complemented, Sylow
171	#575	—	109	1	no	—
172	#568	—	109	1	no	—
173	#568	—	109	1	no	—
174	#444	—	109	2	no	—
175	#444	—	109	2	no	—
176	#443	—	109	2	no	—
177	#443	—	109	2	no	—
178	#430	—	109	1	no	—
179	C₂×#104	—	109	1	no	—
180	C₂×#104	—	109	1	no	—
181	C₄×M₁₆	—	109	1	no	—
182	C₂×#41	—	109	1	no	—
183	C₂×#41	—	109	1	no	—
184	#137	—	109	1	no	—
185	#137	—	109	2	no	—
186	#137	—	109	1	no	—
187	AGL(1,ℤ/8)	—	109	1	no	—
188	AGL(1,ℤ/8)	—	109	2	no	—
189	AGL(1,ℤ/8)	#89137	1	1	yes	characteristic, complemented
190	#135	—	109	2	no	—
191	#135	—	109	2	no	—
192	#131	—	109	2	no	—
193	#131	—	109	2	no	—
194	#131	—	109	2	no	—
195	#131	—	109	2	no	—
196	#131	—	218	1	no	—
197	#131	—	218	1	no	—
198	#131	—	218	2	no	—
199	C₂×M₁₆	—	109	1	no	—
200	C₂×M₁₆	—	109	1	no	—
201	C₂×M₁₆	—	109	1	no	—
202	C₂×M₁₆	—	109	1	no	—
203	C₂×M₁₆	—	218	2	no	—
204	C₂×M₁₆	—	218	1	no	—
205	C₂×M₁₆	—	218	1	no	—
206	C₂×C₄²	—	109	1	no	—
207	#108	—	109	2	no	—
208	#108	—	109	2	no	—
209	#104	—	109	2	no	—
210	#104	—	109	1	no	—
211	#104	—	109	1	no	—
212	#104	—	109	2	no	—
213	#104	—	109	1	no	—
214	#104	—	109	1	no	—
215	#97	—	109	2	no	—
216	C₄×C₈	—	109	2	no	—
217	#41	—	109	1	no	—
218	#41	—	109	2	no	—
219	#41	—	109	2	no	—
220	#41	—	109	1	no	—
221	#41	—	109	1	no	—
222	#41	—	218	1	no	—
223	#41	—	218	1	no	—
224	#41	—	218	1	no	—
225	#41	—	218	1	no	—
226	#41	C₂×#89137	1	1	yes	characteristic
227	C₂×Q₈	—	109	1	no	—
228	C₂×Q₈	—	109	1	no	—
229	C₂×D₄	—	109	1	no	—
230	C₂×D₄	—	218	1	no	—
231	C₂×D₄	—	218	1	no	—
232	C₂×D₄	C₂×#89137	1	1	yes	characteristic
233	C₂²×C₄	—	109	1	no	—
234	C₂²×C₄	—	218	1	no	—
235	C₂²×C₄	—	218	1	no	—
236	C₂²×C₄	—	218	1	no	—
237	Q₁₆	—	109	2	no	—
238	Q₁₆	—	109	2	no	—
239	SD₁₆	—	109	2	no	—
240	SD₁₆	—	109	2	no	—
241	SD₁₆	—	109	2	no	—
242	SD₁₆	C₂×#89137	1	2	yes	complemented
243	D₈	—	109	2	no	—
244	D₈	C₂×#89137	1	2	yes	complemented
245	M₁₆	—	109	2	no	—
246	M₁₆	—	109	1	no	—
247	M₁₆	—	109	1	no	—
248	M₁₆	—	109	1	no	—
249	M₁₆	—	109	2	no	—
250	M₁₆	—	109	2	no	—
251	M₁₆	—	109	1	no	—
252	M₁₆	—	109	1	no	—
253	M₁₆	—	218	2	no	—
254	M₁₆	—	218	2	no	—
255	M₁₆	—	218	2	no	—
256	M₁₆	—	218	2	no	—
257	M₁₆	—	218	1	no	—
258	M₁₆	—	218	1	no	—
259	M₁₆	C₂×#89137	1	1	yes	characteristic
260	C₂×C₈	—	109	2	no	—
261	C₂×C₈	—	109	2	no	—
262	C₂×C₈	—	218	2	no	—
263	C₂×C₈	—	218	2	no	—
264	C₂×C₈	—	218	2	no	—
265	C₂×C₈	—	218	2	no	—
266	C₂×C₈	—	218	1	no	—
267	C₂×C₈	—	218	1	no	—
268	C₄²	—	109	1	no	—
269	C₄²	—	109	1	no	—
270	C₄²	—	218	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	109	1	no	—
273	C₂³	—	218	1	no	—
274	Q₈	—	109	1	no	—
275	Q₈	—	109	2	no	—
276	Q₈	—	218	1	no	—
277	Q₈	—	218	1	no	—
278	Q₈	C₂²×#89137	1	1	yes	characteristic
279	D₄	—	2	1	no	—
280	D₄	—	2	1	no	—
281	D₄	—	109	1	no	—
282	D₄	—	218	1	no	—
283	D₄	—	218	1	no	—
284	D₄	—	218	1	no	—
285	D₄	—	218	2	no	—
286	D₄	—	218	2	no	—
287	D₄	—	218	1	no	—
288	D₄	C₂²×#89137	1	2	yes	—
289	D₄	C₂²×#89137	1	1	yes	characteristic
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	109	1	no	—
292	C₂×C₄	—	109	1	no	—
293	C₂×C₄	—	109	1	no	—
294	C₂×C₄	—	109	1	no	—
295	C₂×C₄	—	109	1	no	—
296	C₂×C₄	—	436	2	no	—
297	C₂×C₄	—	218	1	no	—
298	C₂×C₄	—	218	1	no	—
299	C₂×C₄	—	218	1	no	—
300	C₂×C₄	—	218	2	no	—
301	C₂×C₄	—	218	2	no	—
302	C₂×C₄	—	218	1	no	—
303	C₂×C₄	—	218	1	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
307	C₈	—	109	2	no	—
308	C₈	—	109	2	no	—
309	C₈	—	109	2	no	—
310	C₈	—	218	1	no	—
311	C₈	—	218	1	no	—
312	C₈	—	218	2	no	—
313	C₈	C₂²×#89137	1	2	yes	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	8	1	no	—
317	C₂²	—	872	1	no	—
318	C₂²	—	109	1	no	—
319	C₂²	—	109	1	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	218	1	no	—
322	C₂²	D₄×#89137	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	109	1	no	—
325	C₄	—	109	1	no	—
326	C₄	—	436	1	no	—
327	C₄	—	218	2	no	—
328	C₄	—	218	1	no	—
329	C₄	—	218	2	no	—
330	C₄	D₄×#89137	1	1	yes	characteristic
331	C₄	C₂³×#89137	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	4	2	no	—
334	C₂	—	4	1	no	—
335	C₂	—	436	1	no	—
336	C₂	—	218	1	no	—
337	C₂	—	218	1	no	—
338	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	#2956
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. #20296105	C₂	semidirect
2. #20296104	C₂	semidirect
3. #20296073	C₂	semidirect
4. #20296071	C₂	semidirect
5. #20296069	C₂	semidirect
6. #20296068	C₂	semidirect
7. #20296041	C₂	semidirect
8. D₁₀₉⋊#104	C₂	semidirect
9. C₂×#16812763	C₂	semidirect
10. M₁₆×#89137	C₂	semidirect
12. #16812787	C₂²	semidirect
16. #16812786	C₂²	semidirect
17. #16812786	C₂²	semidirect
18. #16812786	C₂²	semidirect
26. #16812770	C₂²	semidirect
27. #16812769	C₂²	semidirect
28. #16812764	C₂²	semidirect
29. #16812764	C₂²	semidirect
30. #16812763	C₂²	semidirect
31. #16812763	C₂²	semidirect
32. #16812763	C₂²	semidirect
33. #16812763	C₂²	semidirect
34. #16812753	C₂²	semidirect
35. C₈×#89137	C₂²	semidirect
38. #16812732	C₄	semidirect
39. #16812729	C₄	semidirect
40. #16812728	C₄	semidirect
41. #16812725	C₄	semidirect
42. #16812723	C₄	semidirect
46. AGL(1,ℤ/8)×C₁₀₉	C₄	semidirect
87. Dic₄₃₆	C₂×C₄	semidirect
88. #13182019	C₂×C₄	semidirect
89. #13182018	C₂×C₄	semidirect
90. #13182017	C₂×C₄	semidirect
91. #13182017	C₂×C₄	semidirect
92. #13182014	C₂×C₄	semidirect
98. SD₁₆×C₁₀₉	C₂×C₄	semidirect
99. D₈×C₁₀₉	C₂×C₄	semidirect
170. C₁₀₉	#2956	semidirect
189. AGL(1,ℤ/8)	#89137	semidirect
242. SD₁₆	C₂×#89137	semidirect
244. D₈	C₂×#89137	semidirect

Maximal subgroups

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

subgroup	index
#20296105	2
#20296104	2
#20296073	2
#20296071	2
#20296069	2
#20296068	2
#20296041	2
D₁₀₉⋊#104	2
C₂×#16812763	2
M₁₆×#89137	2
#20295822	2
#2956	109