Subgroups — #11737342

Order 896.

456 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#90459	C₂	1	1	yes	maximal, characteristic
2	D₇×#131	C₂	1	2	yes	maximal, complemented
3	C₇×#444	C₂	1	1	yes	maximal, characteristic, complemented
4	#89968	C₂	1	1	yes	maximal, characteristic, complemented
5	#89917	C₂	1	2	yes	maximal, complemented
6	#89667	C₂	1	1	yes	maximal, characteristic, complemented
7	#89665	C₂	1	1	yes	maximal, characteristic, complemented
8	#89596	C₂	1	2	yes	maximal, complemented
9	#89593	C₂	1	2	yes	maximal, complemented
10	#89489	C₂	1	1	yes	maximal, characteristic, complemented
11	#89480	C₂	1	1	yes	maximal, characteristic, complemented
12	D₇×#41	C₂²	1	2	yes	—
13	C₇×#135	C₂²	1	1	yes	characteristic
14	C₇×#131	C₂²	1	2	yes	complemented
15	#6513	C₂²	1	2	yes	—
16	#6511	C₂²	1	2	yes	complemented
17	#6481	C₄	1	2	yes	complemented
18	#6478	C₄	1	2	yes	complemented
19	#6477	C₄	1	2	yes	complemented
20	#6474	C₄	1	2	yes	complemented
21	#6470	—	2	2	no	—
22	#6470	C₂²	1	2	yes	complemented
23	D₇×M₁₆	—	2	2	no	—
24	D₇×M₁₆	C₂²	1	2	yes	—
25	#6467	C₂²	1	1	yes	characteristic
26	#6464	—	2	2	no	—
27	#6464	C₂²	1	2	yes	complemented
28	C₂×#1131	C₂²	1	1	yes	characteristic
29	C₂×C₈×D₇	—	2	2	no	—
30	C₂×C₈×D₇	C₂²	1	1	yes	characteristic
31	#6435	C₂²	1	1	yes	characteristic
32	C₇×#108	C₂²	1	1	yes	characteristic, complemented
33	C₇×#104	C₂²	1	2	yes	complemented
34	C₇×#97	C₂²	1	1	yes	characteristic, complemented
35	#6411	C₂²	1	2	yes	complemented
36	#6399	C₂²	1	2	yes	complemented
37	#6396	C₂²	1	2	yes	complemented
38	#6393	C₂²	1	1	yes	characteristic, complemented
39	#6389	C₂²	1	1	yes	characteristic, complemented
40	C₈×Dic₇	C₂²	1	1	yes	characteristic, complemented
41	#6379	C₂²	1	2	yes	complemented
42	C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#2955	—	7	1	no	maximal, Sylow
44	C₇×#41	C₂³	1	2	yes	—
45	#1161	C₂×C₄	1	2	yes	—
46	Q₈×D₇	C₂×C₄	1	2	yes	—
47	#1159	—	2	2	no	—
48	#1159	C₂×C₄	1	2	yes	—
49	D₄×D₇	—	2	2	no	—
50	D₄×D₇	C₂×C₄	1	2	yes	—
51	#1157	—	2	2	no	—
52	#1157	C₂³	1	2	yes	—
53	C₂×C₄×D₇	—	2	2	no	—
54	C₂×C₄×D₇	C₂³	1	1	yes	characteristic
55	C₇×Q₁₆	C₂×C₄	1	1	yes	characteristic
56	C₇×SD₁₆	C₂×C₄	1	2	yes	—
57	C₇×D₈	C₂×C₄	1	1	yes	characteristic
58	C₇×M₁₆	—	2	2	no	—
59	C₇×M₁₆	C₂³	1	2	yes	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
62	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
63	C₄²×C₇	C₂³	1	1	yes	characteristic
64	#1144	C₂×C₄	1	2	yes	—
65	#1143	C₂×C₄	1	2	yes	—
66	#1142	C₂×C₄	1	2	yes	—
67	#1141	C₂×C₄	1	2	yes	—
68	(C₂×D₇)⋊C₄	—	2	1	no	—
69	Dic₇⋊C₄	—	2	1	no	—
70	C₄×Dic₇	C₂³	1	1	yes	characteristic
71	#1136	—	2	2	no	—
72	#1136	C₂³	1	2	yes	—
73	C₂×C₇⋊C₈	—	2	2	no	—
74	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
75	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
76	Dic₂₈	C₂×C₄	1	1	yes	characteristic
77	D₅₆	C₂×C₄	1	1	yes	characteristic
78	#1132	C₂×C₄	1	2	yes	—
79	#1131	—	2	2	no	—
80	#1131	—	2	2	no	—
81	#1131	—	2	2	no	—
82	#1131	C₂×C₄	1	2	yes	—
83	#1131	C₂×C₄	1	2	yes	—
84	C₈×D₇	—	2	2	no	—
85	C₈×D₇	—	2	2	no	—
86	C₈×D₇	—	2	2	no	—
87	C₈×D₇	D₄	1	2	yes	—
88	C₈×D₇	D₄	1	2	yes	—
89	#575	—	7	1	no	—
90	C₂×#131	—	7	2	no	—
91	#444	—	7	2	no	—
92	#444	—	7	1	no	—
93	#444	D₇	1	1	yes	characteristic, complemented
94	#443	—	7	1	no	—
95	#443	—	7	1	no	—
96	#443	—	7	2	no	—
97	#430	—	7	1	no	—
98	#421	—	7	2	no	—
99	#405	—	7	1	no	—
100	C₂²×D₇	—	2	2	no	—
101	C₂²×D₇	#41	1	1	yes	characteristic
102	C₇×Q₈	C₂²×C₄	1	2	yes	—
103	C₇×D₄	—	2	2	no	—
104	C₇×D₄	C₂²×C₄	1	2	yes	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
108	C₂²⋊D₇	—	2	2	no	—
109	C₂²⋊D₇	—	2	2	no	—
110	C₂²⋊D₇	—	2	2	no	—
111	C₂×Dic₇	—	2	2	no	—
112	C₂×Dic₇	—	2	1	no	—
113	C₂×Dic₇	#41	1	1	yes	characteristic
114	D₂₈	—	2	2	no	—
115	D₂₈	C₂²×C₄	1	2	yes	—
116	C₄×D₇	—	2	2	no	—
117	C₄×D₇	—	2	2	no	—
118	C₄×D₇	—	2	2	no	—
119	C₄×D₇	C₂²×C₄	1	2	yes	—
120	C₄×D₇	C₂×D₄	1	2	yes	—
121	Dic₁₄	—	2	2	no	—
122	Dic₁₄	C₂²×C₄	1	2	yes	—
123	C₇×C₈	—	2	2	no	—
124	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₇×C₈	C₂²×C₄	1	1	yes	characteristic
126	C₇×C₈	C₂×D₄	1	2	yes	—
127	C₇⋊C₈	—	2	2	no	—
128	C₇⋊C₈	C₂²×C₄	1	2	yes	—
129	C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₇⋊C₈	C₂×D₄	1	1	yes	characteristic
131	C₂×#41	—	7	2	no	—
132	#137	—	7	2	no	—
133	#137	—	7	2	no	—
134	AGL(1,ℤ/8)	—	7	2	no	—
135	AGL(1,ℤ/8)	—	7	2	no	—
136	#135	—	7	2	no	—
137	#135	—	7	1	no	—
138	#135	C₂×D₇	1	1	yes	characteristic
139	#131	—	7	2	no	—
140	#131	—	7	2	no	—
141	#131	—	7	2	no	—
142	#131	—	14	2	no	—
143	#131	—	14	2	no	—
144	#131	C₂×D₇	1	2	yes	complemented
145	C₂×M₁₆	—	7	1	no	—
146	C₂×M₁₆	—	7	2	no	—
147	C₂×M₁₆	—	14	2	no	—
148	C₂²×C₈	—	7	1	no	—
149	C₂²×C₈	—	14	2	no	—
150	#117	—	7	1	no	—
151	#108	—	7	2	no	—
152	#108	—	7	1	no	—
153	#108	C₂×D₇	1	1	yes	characteristic, complemented
154	#104	—	7	2	no	—
155	#104	—	7	2	no	—
156	#104	—	7	2	no	—
157	#104	C₂×D₇	1	2	yes	complemented
158	#97	—	7	1	no	—
159	#97	C₂×D₇	1	1	yes	characteristic, complemented
160	C₄×C₈	—	7	1	no	—
161	C₄×C₈	—	7	1	no	—
162	C₂²×C₇	—	2	2	no	—
163	C₂²×C₇	C₂×#41	1	1	yes	characteristic
164	C₂×D₇	—	2	2	no	—
165	C₂×D₇	—	4	1	no	—
166	C₂×D₇	—	4	4	no	—
167	C₂×D₇	C₄×D₄	1	2	yes	—
168	C₄×C₇	—	2	2	no	—
169	C₄×C₇	—	4	1	no	—
170	C₄×C₇	C₂³×C₄	1	1	yes	characteristic
171	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
172	Dic₇	—	2	2	no	—
173	Dic₇	—	4	1	no	—
174	Dic₇	C₄×D₄	1	2	yes	—
175	#41	—	7	2	no	—
176	#41	—	7	2	no	—
177	#41	—	7	2	no	—
178	#41	—	14	2	no	—
179	#41	—	14	2	no	—
180	#41	C₂²×D₇	1	2	yes	—
181	C₂×Q₈	—	7	2	no	—
182	C₂×D₄	—	7	2	no	—
183	C₂×D₄	—	14	2	no	—
184	C₂²×C₄	—	7	1	no	—
185	C₂²×C₄	—	14	2	no	—
186	Q₁₆	—	7	2	no	—
187	Q₁₆	—	7	1	no	—
188	Q₁₆	C₄×D₇	1	1	yes	characteristic
189	SD₁₆	—	7	2	no	—
190	SD₁₆	—	7	2	no	—
191	SD₁₆	—	7	2	no	—
192	SD₁₆	C₄×D₇	1	2	yes	—
193	D₈	—	7	2	no	—
194	D₈	—	7	1	no	—
195	D₈	C₄×D₇	1	1	yes	characteristic
196	M₁₆	—	2	2	no	—
197	M₁₆	—	7	2	no	—
198	M₁₆	—	7	2	no	—
199	M₁₆	—	7	2	no	—
200	M₁₆	—	14	2	no	—
201	M₁₆	—	14	2	no	—
202	M₁₆	—	14	2	no	—
203	M₁₆	—	14	2	no	—
204	M₁₆	C₂²×D₇	1	2	yes	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	7	1	no	—
207	C₂×C₈	—	7	1	no	—
208	C₂×C₈	—	7	2	no	—
209	C₂×C₈	—	7	2	no	—
210	C₂×C₈	—	14	2	no	—
211	C₂×C₈	—	14	2	no	—
212	C₂×C₈	—	14	2	no	—
213	C₂×C₈	—	14	2	no	—
214	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
215	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
216	C₄⋊C₄	—	14	1	no	—
217	C₂²⋊C₄	—	14	1	no	—
218	C₄²	—	7	1	no	—
219	C₄²	C₂²×D₇	1	1	yes	characteristic
220	C₂×C₇	—	2	1	no	—
221	C₂×C₇	—	4	2	no	—
222	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
223	D₇	—	2	2	no	—
224	D₇	—	4	2	no	—
225	C₂³	—	7	1	no	—
226	C₂³	—	14	2	no	—
227	Q₈	—	7	2	no	—
228	Q₈	—	14	2	no	—
229	Q₈	C₂×C₄×D₇	1	2	yes	—
230	D₄	—	2	2	no	—
231	D₄	—	7	2	no	—
232	D₄	—	14	2	no	—
233	D₄	—	14	2	no	—
234	D₄	—	14	2	no	—
235	D₄	—	14	2	no	—
236	D₄	C₂×C₄×D₇	1	2	yes	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	7	1	no	—
240	C₂×C₄	—	7	2	no	—
241	C₂×C₄	—	7	2	no	—
242	C₂×C₄	—	14	2	no	—
243	C₂×C₄	—	14	2	no	—
244	C₂×C₄	—	14	2	no	—
245	C₂×C₄	—	14	2	no	—
246	C₂×C₄	—	14	1	no	—
247	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	7	2	no	—
250	C₈	—	7	1	no	—
251	C₈	—	7	1	no	—
252	C₈	—	14	2	no	—
253	C₈	C₂×C₄×D₇	1	1	yes	characteristic
254	C₈	C₂×C₄×D₇	1	1	yes	characteristic
255	C₈	D₄×D₇	1	2	yes	—
256	C₇	#2955	1	1	yes	characteristic, complemented, Sylow
257	C₂²	—	2	2	no	—
258	C₂²	—	7	2	no	—
259	C₂²	—	14	2	no	—
260	C₂²	—	28	1	no	—
261	C₂²	—	28	4	no	—
262	C₂²	D₇×#41	1	1	yes	characteristic
263	C₄	—	2	2	no	—
264	C₄	—	4	1	no	—
265	C₄	—	7	2	no	—
266	C₄	—	14	2	no	—
267	C₄	—	28	1	no	—
268	C₄	C₂²×C₄×D₇	1	1	yes	characteristic
269	C₄	C₂×D₄×D₇	1	1	yes	characteristic
270	C₂	—	2	1	no	—
271	C₂	—	4	2	no	—
272	C₂	—	14	2	no	—
273	C₂	—	28	2	no	—
274	C₂	C₄×D₄×D₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₇
commutator	C₄×C₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#444	C₂
socle	C₂×C₇	C₂×C₄×D₄
prime radical	#90459	C₂
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#444	D₇
hypocenter	C₇	#2955
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #444

Complements

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

N	H ≅ G/N	type
2. D₇×#131	C₂	semidirect
3. C₇×#444	C₂	semidirect
4. #89968	C₂	semidirect
5. #89917	C₂	semidirect
6. #89667	C₂	semidirect
7. #89665	C₂	semidirect
8. #89596	C₂	semidirect
9. #89593	C₂	semidirect
10. #89489	C₂	semidirect
11. #89480	C₂	semidirect
14. C₇×#131	C₂²	semidirect
16. #6511	C₂²	semidirect
17. #6481	C₄	semidirect
18. #6478	C₄	semidirect
19. #6477	C₄	semidirect
20. #6474	C₄	semidirect
22. #6470	C₂²	semidirect
27. #6464	C₂²	semidirect
32. C₇×#108	C₂²	semidirect
33. C₇×#104	C₂²	semidirect
34. C₇×#97	C₂²	semidirect
35. #6411	C₂²	semidirect
36. #6399	C₂²	semidirect
37. #6396	C₂²	semidirect
38. #6393	C₂²	semidirect
39. #6389	C₂²	semidirect
40. C₈×Dic₇	C₂²	semidirect
41. #6379	C₂²	semidirect
42. C₄×C₇⋊C₈	C₂²	semidirect
93. #444	D₇	semidirect
144. #131	C₂×D₇	semidirect
153. #108	C₂×D₇	semidirect
157. #104	C₂×D₇	semidirect
159. #97	C₂×D₇	semidirect
256. C₇	#2955	semidirect

Maximal subgroups

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

subgroup	index
#90459	2
D₇×#131	2
C₇×#444	2
#89968	2
#89917	2
#89667	2
#89665	2
#89596	2
#89593	2
#89489	2
#89480	2
#2955	7