Subgroups — #28346370

Order 13696.

420 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16812068	C₂	1	1	yes	maximal, characteristic, complemented
2	#20292965	C₂	1	2	yes	maximal, complemented
3	#20292960	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292955	C₂	1	1	yes	maximal, characteristic, complemented
5	#20292948	C₂	1	2	yes	maximal, complemented
6	#20292943	C₂	1	1	yes	maximal, characteristic, complemented
7	#20292938	C₂	1	1	yes	maximal, characteristic, complemented
8	#20292663	C₂	1	2	yes	maximal, complemented
9	#20292623	C₂	1	2	yes	maximal, complemented
10	C₂×C₂²⋊(C₁₀₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#525×C₁₀₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₁₀₇⋊C₈	C₂²	1	2	yes	—
13	#16812068	—	2	1	no	—
14	#16812068	—	2	1	no	—
15	#16812068	C₂²	1	2	yes	—
16	#16812068	C₂²	1	2	yes	—
17	C₂×C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₁₀₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₁₀₇	—	2	1	no	—
20	C₂×C₄×Dic₁₀₇	—	2	1	no	—
21	#16812042	C₂²	1	2	yes	complemented
22	#16812042	C₂²	1	2	yes	complemented
23	#16812040	C₂²	1	2	yes	complemented
24	#16812040	C₂²	1	2	yes	complemented
25	#16812033	C₂²	1	2	yes	complemented
26	#16812033	C₂²	1	2	yes	complemented
27	#16812030	C₂²	1	2	yes	complemented
28	#16812030	C₂²	1	2	yes	complemented
29	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
30	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	1	yes	characteristic
31	C₂²⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
32	C₂×#41×C₁₀₇	C₂²	1	1	yes	characteristic
33	C₂³×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
34	#123×C₁₀₇	—	2	1	no	—
35	C₂²⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
36	C₂²⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
37	#121×C₁₀₇	C₂²	1	1	yes	characteristic
38	#121×C₁₀₇	C₂²	1	1	yes	characteristic
39	#120×C₁₀₇	—	2	1	no	—
40	C₄×D₄×C₁₀₇	—	2	1	no	—
41	C₄×D₄×C₁₀₇	—	2	1	no	—
42	#117×C₁₀₇	C₂²	1	1	yes	characteristic
43	C₂²×Dic₁₀₇	—	2	1	no	—
44	C₂²×Dic₁₀₇	—	2	1	no	—
45	C₂×C₁₀₇⋊C₈	—	2	4	no	—
46	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
47	C₂×C₁₀₇⋊C₈	C₂³	1	2	yes	—
48	C₄⋊Dic₁₀₇	—	2	1	no	—
49	C₄⋊Dic₁₀₇	—	2	1	no	—
50	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
51	C₄⋊Dic₁₀₇	C₂³	1	2	yes	—
52	C₂²⋊Dic₁₀₇	—	2	2	no	—
53	C₂²⋊Dic₁₀₇	—	2	2	no	—
54	C₄×Dic₁₀₇	—	2	2	no	—
55	C₄×Dic₁₀₇	—	2	2	no	—
56	C₂⁴×C₁₀₇	D₄	1	1	yes	characteristic
57	#41×C₁₀₇	—	2	4	no	—
58	C₂×Q₈×C₁₀₇	C₂³	1	1	yes	characteristic
59	C₂×D₄×C₁₀₇	—	2	1	no	—
60	C₂×D₄×C₁₀₇	—	2	1	no	—
61	C₂×D₄×C₁₀₇	—	2	1	no	—
62	C₂×D₄×C₁₀₇	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₀₇	—	2	4	no	—
64	C₂²×C₄×C₁₀₇	—	2	1	no	—
65	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
66	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
67	C₂²×C₄×C₁₀₇	D₄	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₁₀₇	—	2	1	no	—
72	C₄⋊C₄×C₁₀₇	—	2	1	no	—
73	C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₁₀₇	C₂³	1	1	yes	characteristic
75	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
76	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
77	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
78	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
79	C₂²⋊C₄×C₁₀₇	—	2	1	no	—
80	C₄²×C₁₀₇	—	2	1	no	—
81	C₂³×C₁₀₇	—	2	2	no	—
82	C₂³×C₁₀₇	—	2	2	no	—
83	C₂³×C₁₀₇	—	2	1	no	—
84	C₂³×C₁₀₇	—	4	1	no	—
85	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
86	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
87	C₂³×C₁₀₇	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₁₀₇	—	2	2	no	—
89	D₄×C₁₀₇	—	2	2	no	—
90	D₄×C₁₀₇	—	4	2	no	—
91	D₄×C₁₀₇	—	4	2	no	—
92	D₄×C₁₀₇	—	4	2	no	—
93	C₂×C₄×C₁₀₇	—	2	2	no	—
94	C₂×C₄×C₁₀₇	—	2	2	no	—
95	C₂×C₄×C₁₀₇	—	2	2	no	—
96	C₂×C₄×C₁₀₇	—	2	2	no	—
97	C₂×C₄×C₁₀₇	—	2	2	no	—
98	C₂×C₄×C₁₀₇	—	2	1	no	—
99	C₂×C₄×C₁₀₇	—	2	1	no	—
100	C₂×C₄×C₁₀₇	—	2	1	no	—
101	C₂×C₄×C₁₀₇	—	4	2	no	—
102	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₀₇	#41	1	4	yes	—
106	C₂×C₄×C₁₀₇	C₂⁴	1	1	yes	characteristic
107	C₂×Dic₁₀₇	—	2	2	no	—
108	C₂×Dic₁₀₇	—	2	2	no	—
109	C₂×Dic₁₀₇	—	4	2	no	—
110	C₂×Dic₁₀₇	—	4	2	no	—
111	C₁₀₇⋊C₈	—	2	2	no	—
112	C₁₀₇⋊C₈	—	2	2	no	—
113	C₂²×C₁₀₇	—	2	1	no	—
114	C₂²×C₁₀₇	—	2	1	no	—
115	C₂²×C₁₀₇	—	2	1	no	—
116	C₂²×C₁₀₇	—	2	1	no	—
117	C₂²×C₁₀₇	—	2	1	no	—
118	C₂²×C₁₀₇	—	2	1	no	—
119	C₂²×C₁₀₇	—	2	1	no	—
120	C₂²×C₁₀₇	—	4	4	no	—
121	C₂²×C₁₀₇	—	4	1	no	—
122	C₂²×C₁₀₇	—	4	1	no	—
123	C₂²×C₁₀₇	—	4	1	no	—
124	C₂²×C₁₀₇	#123	1	2	yes	—
125	C₂²×C₁₀₇	#135	1	2	yes	—
126	C₂²×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₀₇	—	2	1	no	—
128	C₄×C₁₀₇	—	2	1	no	—
129	C₄×C₁₀₇	—	4	1	no	—
130	C₄×C₁₀₇	—	4	1	no	—
131	C₄×C₁₀₇	—	4	1	no	—
132	C₄×C₁₀₇	#123	1	2	yes	—
133	C₄×C₁₀₇	C₂×#41	1	2	yes	—
134	Dic₁₀₇	—	4	2	no	—
135	Dic₁₀₇	—	4	2	no	—
136	C₂×C₁₀₇	—	2	2	no	—
137	C₂×C₁₀₇	—	2	2	no	—
138	C₂×C₁₀₇	—	4	1	no	—
139	C₂×C₁₀₇	—	8	1	no	—
140	C₂×C₁₀₇	C₂×#123	1	1	yes	characteristic
141	C₂×C₁₀₇	C₂×#135	1	1	yes	characteristic
142	C₂×C₁₀₇	#575	1	1	yes	characteristic
143	#3091	—	107	1	no	maximal, Sylow
144	C₁₀₇	#3091	1	1	yes	characteristic, complemented, Sylow
145	#525	D₁₀₇	1	1	yes	characteristic
146	C₂×#117	—	107	1	no	—
147	#485	—	107	2	no	—
148	#484	—	107	1	no	—
149	#483	—	107	1	no	—
150	#482	—	107	2	no	—
151	#481	—	107	1	no	—
152	#480	—	107	1	no	—
153	#427	—	107	2	no	—
154	#416	—	107	2	no	—
155	C₂×C₂²⋊C₈	—	107	1	no	—
156	C₂×#41	C₂×D₁₀₇	1	1	yes	characteristic
157	C₂³×C₄	C₂×D₁₀₇	1	1	yes	characteristic
158	C₂²×C₈	—	107	2	no	—
159	#123	—	2	1	no	—
160	C₂²⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
161	C₂²⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
162	#121	C₂×D₁₀₇	1	1	yes	characteristic
163	#121	C₂×D₁₀₇	1	1	yes	characteristic
164	#120	—	2	1	no	—
165	C₄×D₄	—	2	1	no	—
166	C₄×D₄	—	2	1	no	—
167	#117	—	107	2	no	—
168	#117	—	107	2	no	—
169	#117	—	214	1	no	—
170	#117	—	214	1	no	—
171	#117	C₂×D₁₀₇	1	1	yes	characteristic
172	C₂×C₄⋊C₄	—	107	1	no	—
173	C₂×C₄⋊C₄	—	107	1	no	—
174	C₂×C₂²⋊C₄	—	214	1	no	—
175	C₂×C₄²	—	214	1	no	—
176	#107	—	107	2	no	—
177	#107	—	107	2	no	—
178	#106	—	107	2	no	—
179	#106	—	107	2	no	—
180	#103	—	107	2	no	—
181	#103	—	107	2	no	—
182	#102	—	107	2	no	—
183	#102	—	107	2	no	—
184	C₂²⋊C₈	—	107	1	no	—
185	C₂²⋊C₈	—	107	1	no	—
186	C₂²⋊C₈	—	107	2	no	—
187	C₂⁴	C₂²⋊D₁₀₇	1	1	yes	characteristic
188	#41	—	2	4	no	—
189	C₂×Q₈	C₂²×D₁₀₇	1	1	yes	characteristic
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	C₂²×D₁₀₇	1	1	yes	characteristic
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	1	no	—
196	C₂²×C₄	—	214	1	no	—
197	C₂²×C₄	—	214	1	no	—
198	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
199	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
200	C₂²×C₄	C₂²⋊D₁₀₇	1	1	yes	characteristic
201	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
202	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
204	C₂×C₈	—	107	2	no	—
205	C₂×C₈	—	107	2	no	—
206	C₂×C₈	—	214	4	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	107	2	no	—
210	C₄⋊C₄	—	107	2	no	—
211	C₄⋊C₄	—	214	1	no	—
212	C₄⋊C₄	—	214	1	no	—
213	C₄⋊C₄	C₂²×D₁₀₇	1	1	yes	characteristic
214	C₄⋊C₄	C₂²×D₁₀₇	1	1	yes	characteristic
215	C₂²⋊C₄	—	2	1	no	—
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	214	2	no	—
221	C₂²⋊C₄	—	214	2	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	214	2	no	—
224	C₄²	—	214	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
230	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
231	C₂³	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
232	Q₈	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	4	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	428	2	no	—
247	C₂×C₄	—	428	2	no	—
248	C₂×C₄	—	214	2	no	—
249	C₂×C₄	—	214	2	no	—
250	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
252	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
253	C₂×C₄	#13133817	1	4	yes	—
254	C₂×C₄	C₂³×D₁₀₇	1	1	yes	characteristic
255	C₈	—	214	2	no	—
256	C₈	—	214	2	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	2	1	no	—
259	C₂²	—	2	1	no	—
260	C₂²	—	2	1	no	—
261	C₂²	—	2	1	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	4	4	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#16812090	1	2	yes	—
269	C₂²	#16812129	1	2	yes	—
270	C₂²	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
271	C₄	—	2	1	no	—
272	C₄	—	2	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	4	1	no	—
275	C₄	—	4	1	no	—
276	C₄	—	428	2	no	—
277	C₄	—	428	2	no	—
278	C₄	#16812090	1	2	yes	—
279	C₄	C₂×#13133817	1	2	yes	—
280	C₂	—	2	2	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	1	no	—
283	C₂	—	8	1	no	—
284	C₂	C₂×#16812090	1	1	yes	characteristic
285	C₂	C₂×#16812129	1	1	yes	characteristic
286	C₂	#20293425	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₇
commutator, cosocle	C₂×C₄×C₁₀₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₇
Fitting, prime radical	#525×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂²×D₄
hypercenter	#525	D₁₀₇
hypocenter	C₁₀₇	#3091
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#16812068	C₂	semidirect
2. #20292965	C₂	semidirect
3. #20292960	C₂	semidirect
4. #20292955	C₂	semidirect
5. #20292948	C₂	semidirect
6. #20292943	C₂	semidirect
7. #20292938	C₂	semidirect
8. #20292663	C₂	semidirect
9. #20292623	C₂	semidirect
10. C₂×C₂²⋊(C₁₀₇⋊C₈)	C₂	semidirect
21. #16812042	C₂²	semidirect
22. #16812042	C₂²	semidirect
23. #16812040	C₂²	semidirect
24. #16812040	C₂²	semidirect
25. #16812033	C₂²	semidirect
26. #16812033	C₂²	semidirect
27. #16812030	C₂²	semidirect
28. #16812030	C₂²	semidirect
144. C₁₀₇	#3091	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16812068	2
#20292965	2
#20292960	2
#20292955	2
#20292948	2
#20292943	2
#20292938	2
#20292663	2
#20292623	2
C₂×C₂²⋊(C₁₀₇⋊C₈)	2
#525×C₁₀₇	2
#3091	107