Subgroups — #11740593

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#529	C₂	1	1	yes	maximal, characteristic, complemented
2	#90168	C₂	1	1	yes	maximal, characteristic, complemented
3	#89838	C₂	1	2	yes	maximal, complemented
4	#89829	C₂	1	2	yes	maximal, complemented
5	#89820	C₂	1	2	yes	maximal, complemented
6	#89810	C₂	1	2	yes	maximal, complemented
7	#89799	C₂	1	1	yes	maximal, characteristic, complemented
8	#89793	C₂	1	1	yes	maximal, characteristic, complemented
9	#89787	C₂	1	1	yes	maximal, characteristic, complemented
10	#89780	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄⋊(C₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#1157	—	2	1	no	—
13	C₇×#126	—	2	1	no	—
14	C₇×#125	C₂²	1	1	yes	characteristic
15	C₇×#124	C₂²	1	1	yes	characteristic
16	C₇×#123	—	2	1	no	—
17	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₇×#121	C₂²	1	1	yes	characteristic
19	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
20	C₄×C₇×D₄	—	2	1	no	—
21	C₄×C₇×D₄	C₂²	1	1	yes	characteristic
22	C₇×#117	—	2	1	no	—
23	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
24	C₂×#1143	C₂²	1	2	yes	—
25	C₂×#1141	C₂²	1	2	yes	—
26	#6493	—	2	1	no	—
27	#6493	C₂²	1	2	yes	—
28	C₂²×C₇⋊C₈	C₂²	1	2	yes	—
29	#6438	—	2	1	no	—
30	#6437	C₂²	1	1	yes	characteristic
31	C₄×D₂₈	—	2	1	no	—
32	C₄×D₂₈	C₂²	1	2	yes	—
33	#6432	C₂²	1	1	yes	characteristic
34	#6409	C₂²	1	2	yes	—
35	#6406	C₂²	1	2	yes	—
36	#6383	C₂²	1	2	yes	—
37	#6383	C₂²	1	2	yes	—
38	#6382	C₂²	1	2	yes	—
39	#6381	C₂²	1	2	yes	—
40	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
42	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
43	#3247	—	7	1	no	maximal, Sylow
44	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
45	C₂×C₇×D₄	—	2	1	no	—
46	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₇	—	2	2	no	—
52	#1157	—	4	4	no	—
53	C₂×D₂₈	—	2	1	no	—
54	C₂×D₂₈	—	2	1	no	—
55	C₂×D₂₈	C₂³	1	2	yes	—
56	C₂×C₄×D₇	—	2	2	no	—
57	C₂×Dic₁₄	—	2	1	no	—
58	C₇×C₄⋊C₄	—	2	1	no	—
59	C₇×C₄⋊C₄	—	2	1	no	—
60	C₇×C₄⋊C₄	—	2	1	no	—
61	C₇×C₄⋊C₄	—	2	1	no	—
62	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₇×C₂²⋊C₄	—	2	1	no	—
65	C₇×C₂²⋊C₄	—	2	1	no	—
66	C₇×C₂²⋊C₄	—	2	1	no	—
67	C₇×C₂²⋊C₄	—	2	1	no	—
68	C₇×C₂²⋊C₄	—	2	1	no	—
69	C₄²×C₇	—	2	1	no	—
70	C₄²×C₇	D₄	1	1	yes	characteristic
71	C₄²×C₇	D₄	1	1	yes	characteristic
72	C₄²×C₇	C₂³	1	1	yes	characteristic
73	C₄²×C₇	C₂³	1	1	yes	characteristic
74	#1143	—	4	2	no	—
75	#1141	—	4	2	no	—
76	(C₂×D₇)⋊C₄	—	2	2	no	—
77	(C₂×D₇)⋊C₄	—	2	2	no	—
78	C₄⋊Dic₇	—	2	1	no	—
79	C₄⋊Dic₇	C₂³	1	2	yes	—
80	C₂×C₇⋊C₈	—	4	2	no	—
81	C₂×C₇⋊C₈	C₂³	1	2	yes	—
82	C₂×C₇⋊C₈	C₂³	1	2	yes	—
83	#532	—	7	1	no	—
84	#529	D₇	1	1	yes	characteristic, complemented
85	#478	—	7	2	no	—
86	#477	—	7	1	no	—
87	#476	—	7	1	no	—
88	#466	—	7	2	no	—
89	#465	—	7	2	no	—
90	#463	—	7	2	no	—
91	#460	—	7	1	no	—
92	#459	—	7	1	no	—
93	C₂×C₄⋊C₈	—	7	1	no	—
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×D₇	—	2	2	no	—
97	C₇×Q₈	—	2	1	no	—
98	C₇×Q₈	—	2	1	no	—
99	C₇×D₄	—	2	1	no	—
100	C₇×D₄	—	2	1	no	—
101	C₇×D₄	—	4	2	no	—
102	C₂×C₄×C₇	—	2	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	1	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	4	2	no	—
110	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₇	—	4	4	no	—
117	C₂×Dic₇	—	2	2	no	—
118	D₂₈	—	2	2	no	—
119	D₂₈	—	2	2	no	—
120	D₂₈	—	4	2	no	—
121	D₂₈	—	4	2	no	—
122	C₄×D₇	—	4	4	no	—
123	Dic₁₄	—	4	2	no	—
124	C₇⋊C₈	—	2	2	no	—
125	C₇⋊C₈	—	2	2	no	—
126	C₂×#41	—	14	1	no	—
127	C₂×SD₁₆	—	7	2	no	—
128	C₂×D₈	—	7	2	no	—
129	C₂²×C₈	—	7	2	no	—
130	C₄⋊Q₈	—	7	1	no	—
131	#127	—	7	1	no	—
132	#126	—	2	1	no	—
133	#125	C₂×D₇	1	1	yes	characteristic
134	#124	—	14	1	no	—
135	#124	C₂×D₇	1	1	yes	characteristic
136	#123	—	2	1	no	—
137	C₂²⋊Q₈	C₂×D₇	1	1	yes	characteristic
138	#121	—	7	2	no	—
139	#121	—	14	1	no	—
140	#121	C₂×D₇	1	1	yes	characteristic
141	C₄×Q₈	C₂×D₇	1	1	yes	characteristic
142	C₄×D₄	—	2	1	no	—
143	C₄×D₄	—	7	2	no	—
144	C₄×D₄	—	14	1	no	—
145	C₄×D₄	C₂×D₇	1	1	yes	characteristic
146	#117	—	2	1	no	—
147	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
148	#107	—	7	2	no	—
149	#106	—	7	2	no	—
150	C₄⋊C₈	—	7	1	no	—
151	C₄⋊C₈	—	7	1	no	—
152	C₄⋊C₈	—	7	2	no	—
153	#103	—	7	2	no	—
154	#102	—	7	2	no	—
155	#102	—	7	2	no	—
156	#102	—	7	2	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	4	1	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	—	4	1	no	—
162	C₂²×C₇	AGL(1,ℤ/8)	1	2	yes	—
163	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₂×D₇	—	4	2	no	—
165	C₂×D₇	—	4	2	no	—
166	C₂×D₇	—	4	2	no	—
167	C₄×C₇	—	2	1	no	—
168	C₄×C₇	—	2	1	no	—
169	C₄×C₇	—	2	1	no	—
170	C₄×C₇	—	2	1	no	—
171	C₄×C₇	—	4	1	no	—
172	C₄×C₇	—	4	1	no	—
173	C₄×C₇	—	4	1	no	—
174	C₄×C₇	#135	1	2	yes	—
175	C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
176	C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
177	Dic₇	—	4	2	no	—
178	#41	—	28	4	no	—
179	C₂×Q₈	—	14	1	no	—
180	C₂×Q₈	C₂²×D₇	1	1	yes	characteristic
181	C₂×D₄	—	2	1	no	—
182	C₂×D₄	—	7	2	no	—
183	C₂×D₄	—	14	2	no	—
184	C₂×D₄	—	14	1	no	—
185	C₂×D₄	—	14	1	no	—
186	C₂×D₄	C₂²×D₇	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	14	2	no	—
189	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²⋊D₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
192	SD₁₆	—	28	2	no	—
193	D₈	—	28	2	no	—
194	C₂×C₈	—	7	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₂×C₈	—	28	2	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	7	2	no	—
202	C₄⋊C₄	—	14	1	no	—
203	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
204	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	14	2	no	—
211	C₂²⋊C₄	—	14	2	no	—
212	C₄²	—	2	1	no	—
213	C₄²	C₂²⋊D₇	1	1	yes	characteristic
214	C₄²	C₂²⋊D₇	1	1	yes	characteristic
215	C₄²	C₂²×D₇	1	1	yes	characteristic
216	C₄²	C₂²×D₇	1	1	yes	characteristic
217	C₂×C₇	—	2	2	no	—
218	C₂×C₇	—	8	1	no	—
219	C₂×C₇	#537	1	1	yes	characteristic
220	C₂×C₇	C₂×#135	1	1	yes	characteristic
221	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
222	D₇	—	8	2	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	14	2	no	—
225	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
226	Q₈	—	2	1	no	—
227	Q₈	—	2	1	no	—
228	Q₈	—	28	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	1	no	—
231	D₄	—	4	2	no	—
232	D₄	—	14	2	no	—
233	D₄	—	14	2	no	—
234	D₄	—	28	2	no	—
235	D₄	—	28	4	no	—
236	D₄	—	28	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	14	2	no	—
246	C₂×C₄	—	28	4	no	—
247	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
251	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
253	C₈	—	14	2	no	—
254	C₈	—	14	2	no	—
255	C₇	#3247	1	1	yes	characteristic, complemented, Sylow
256	C₂²	—	2	1	no	—
257	C₂²	—	2	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	28	2	no	—
262	C₂²	—	28	2	no	—
263	C₂²	—	28	2	no	—
264	C₂²	#6512	1	2	yes	—
265	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	2	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	1	no	—
273	C₄	—	28	2	no	—
274	C₄	#6513	1	2	yes	—
275	C₄	#6553	1	1	yes	characteristic
276	C₄	#6554	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	8	1	no	—
279	C₂	—	56	2	no	—
280	C₂	C₂×#6512	1	1	yes	characteristic
281	C₂	C₂×#6513	1	1	yes	characteristic
282	C₂	#90524	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#529	C₂
socle	C₂²×C₇	C₂²×D₄
hypercenter	#529	D₇
hypocenter	C₇	#3247
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #529

Complements

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

N	H ≅ G/N	type
1. C₇×#529	C₂	semidirect
2. #90168	C₂	semidirect
3. #89838	C₂	semidirect
4. #89829	C₂	semidirect
5. #89820	C₂	semidirect
6. #89810	C₂	semidirect
7. #89799	C₂	semidirect
8. #89793	C₂	semidirect
9. #89787	C₂	semidirect
10. #89780	C₂	semidirect
11. C₂×C₄⋊(C₇⋊C₈)	C₂	semidirect
84. #529	D₇	semidirect
255. C₇	#3247	semidirect

Maximal subgroups

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

subgroup	index
C₇×#529	2
#90168	2
#89838	2
#89829	2
#89820	2
#89810	2
#89799	2
#89793	2
#89787	2
#89780	2
C₂×C₄⋊(C₇⋊C₈)	2
#3247	7