Subgroups — #11740586

Order 896.

454 conjugacy classes of subgroups.

The subgroup lattice has 282 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	C₂×C₄×D₂₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#89838	C₂	1	2	yes	maximal, complemented
4	#89829	C₂	1	2	yes	maximal, complemented
5	#89818	C₂	1	1	yes	maximal, characteristic, complemented
6	#89816	C₂	1	1	yes	maximal, characteristic, complemented
7	#89808	C₂	1	1	yes	maximal, characteristic
8	#89806	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#1143	C₂	1	2	yes	maximal, complemented
10	C₄×#1141	C₂	1	2	yes	maximal, complemented
11	#89697	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₂₈	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₇	—	2	1	no	—
14	C₇×#126	—	2	1	no	—
15	C₇×#125	C₂²	1	1	yes	characteristic, complemented
16	C₇×#124	C₂²	1	1	yes	characteristic, complemented
17	C₇×#123	—	2	1	no	—
18	C₇×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
19	C₇×#121	C₂²	1	1	yes	characteristic
20	C₄×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
21	C₄×C₇×D₄	—	2	1	no	—
22	C₄×C₇×D₄	C₂²	1	1	yes	characteristic, complemented
23	C₇×#117	—	2	1	no	—
24	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
25	C₂×#1143	C₂²	1	2	yes	—
26	C₂×#1141	C₂²	1	2	yes	—
27	C₂×(C₂×D₇)⋊C₄	—	2	1	no	—
28	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
29	C₄×D₂₈	—	2	1	no	—
30	C₄×D₂₈	—	2	1	no	—
31	C₄×D₂₈	C₂²	1	2	yes	—
32	C₄×D₂₈	C₂²	1	2	yes	—
33	#6409	C₂²	1	2	yes	complemented
34	#6406	C₂²	1	2	yes	complemented
35	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
36	C₂²⋊(C₇⋊C₈)	C₂²	1	1	yes	characteristic
37	#6383	C₂²	1	2	yes	—
38	#6383	C₂²	1	2	yes	—
39	#6382	C₂²	1	2	yes	complemented
40	#6381	C₂²	1	2	yes	complemented
41	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	complemented
42	C₄×C₇⋊C₈	C₂²	1	2	yes	complemented
43	#3105	—	7	1	no	maximal, Sylow
44	C₂³×D₇	—	2	1	no	—
45	C₂×C₇×Q₈	C₂³	1	1	yes	characteristic
46	C₂×C₇×D₄	—	2	1	no	—
47	C₂×C₇×D₄	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	1	no	—
49	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
52	C₂²×Dic₇	—	2	1	no	—
53	C₂×D₂₈	—	2	1	no	—
54	C₂×D₂₈	—	2	4	no	—
55	C₂×D₂₈	C₂³	1	2	yes	—
56	C₂×C₄×D₇	—	2	2	no	—
57	C₂×C₄×D₇	—	2	2	no	—
58	C₂×C₄×D₇	—	4	4	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	—	2	1	no	—
63	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
64	C₇×C₄⋊C₄	C₂³	1	1	yes	characteristic
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	C₇×C₂²⋊C₄	—	2	1	no	—
70	C₄²×C₇	—	2	1	no	—
71	C₄²×C₇	D₄	1	1	yes	characteristic
72	C₄²×C₇	D₄	1	1	yes	characteristic
73	C₄²×C₇	C₂³	1	1	yes	characteristic
74	C₄²×C₇	C₂³	1	1	yes	characteristic
75	#1143	—	2	4	no	—
76	#1141	—	2	4	no	—
77	(C₂×D₇)⋊C₄	—	2	2	no	—
78	(C₂×D₇)⋊C₄	—	2	2	no	—
79	C₄⋊Dic₇	—	2	1	no	—
80	C₄⋊Dic₇	C₂³	1	2	yes	—
81	C₂×C₇⋊C₈	C₂³	1	2	yes	—
82	C₂×C₇⋊C₈	C₂³	1	2	yes	—
83	#529	D₇	1	1	yes	characteristic, complemented
84	C₂×C₄×D₄	—	7	1	no	—
85	#484	—	7	1	no	—
86	#481	—	7	1	no	—
87	#478	—	7	2	no	—
88	#463	—	7	2	no	—
89	#450	—	7	1	no	—
90	#447	—	7	1	no	—
91	C₄×SD₁₆	—	7	2	no	—
92	C₄×D₈	—	7	2	no	—
93	#431	—	7	1	no	—
94	C₂³×C₇	—	2	1	no	—
95	C₂³×C₇	C₂×D₄	1	1	yes	characteristic
96	C₂²×D₇	—	2	4	no	—
97	C₂²×D₇	—	2	2	no	—
98	C₂²×D₇	—	4	2	no	—
99	C₂²×D₇	—	4	2	no	—
100	C₇×Q₈	—	2	2	no	—
101	C₇×D₄	—	2	2	no	—
102	C₇×D₄	—	4	2	no	—
103	C₂×C₄×C₇	—	2	2	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	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	C₂×Dic₇	—	2	2	no	—
117	C₂×Dic₇	—	4	2	no	—
118	D₂₈	—	2	4	no	—
119	D₂₈	—	2	2	no	—
120	D₂₈	#41	1	4	yes	—
121	C₄×D₇	—	4	4	no	—
122	C₄×D₇	—	4	4	no	—
123	C₇⋊C₈	—	2	2	no	—
124	C₇⋊C₈	—	2	2	no	—
125	C₂²×D₄	—	7	1	no	—
126	C₂³×C₄	—	14	1	no	—
127	C₂×SD₁₆	—	7	2	no	—
128	C₂×D₈	—	7	2	no	—
129	#126	—	2	1	no	—
130	#125	C₂×D₇	1	1	yes	characteristic, complemented
131	#124	C₂×D₇	1	1	yes	characteristic, complemented
132	#123	—	2	1	no	—
133	C₂²⋊Q₈	C₂×D₇	1	1	yes	characteristic
134	#121	C₂×D₇	1	1	yes	characteristic
135	C₄×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
136	C₄×D₄	—	2	1	no	—
137	C₄×D₄	—	7	2	no	—
138	C₄×D₄	—	7	2	no	—
139	C₄×D₄	—	14	1	no	—
140	C₄×D₄	—	14	1	no	—
141	C₄×D₄	C₂×D₇	1	1	yes	characteristic, complemented
142	#117	—	2	1	no	—
143	C₂×C₄⋊C₄	—	7	1	no	—
144	C₂×C₂²⋊C₄	—	14	1	no	—
145	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
146	#107	—	7	2	no	—
147	#106	—	7	2	no	—
148	C₄⋊C₈	—	7	2	no	—
149	#103	—	7	2	no	—
150	#102	—	7	2	no	—
151	#102	—	7	2	no	—
152	#102	—	7	2	no	—
153	C₂²⋊C₈	—	7	1	no	—
154	C₂²⋊C₈	—	7	1	no	—
155	C₄×C₈	—	7	2	no	—
156	C₂²×C₇	—	2	1	no	—
157	C₂²×C₇	—	2	1	no	—
158	C₂²×C₇	—	4	1	no	—
159	C₂²×C₇	—	4	1	no	—
160	C₂²×C₇	—	4	1	no	—
161	C₂²×C₇	#135	1	2	yes	—
162	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₇	—	2	4	no	—
164	C₂×D₇	—	4	8	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	AGL(1,ℤ/8)	1	2	yes	—
175	C₄×C₇	C₂×#41	1	2	yes	—
176	Dic₇	—	4	2	no	—
177	C₂⁴	—	14	1	no	—
178	C₂×Q₈	C₂²×D₇	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	—	7	2	no	—
181	C₂×D₄	—	14	4	no	—
182	C₂×D₄	—	14	1	no	—
183	C₂×D₄	C₂²×D₇	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	14	1	no	—
186	C₂²×C₄	—	14	2	no	—
187	C₂²×C₄	—	14	2	no	—
188	C₂²×C₄	—	28	4	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	—	14	4	no	—
193	D₈	—	14	4	no	—
194	C₂×C₈	—	7	2	no	—
195	C₂×C₈	—	7	2	no	—
196	C₄⋊C₄	—	2	1	no	—
197	C₄⋊C₄	—	2	1	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	7	2	no	—
201	C₄⋊C₄	—	14	1	no	—
202	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
203	C₄⋊C₄	C₂²×D₇	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
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	—	14	2	no	—
210	C₂²⋊C₄	—	14	2	no	—
211	C₄²	—	2	1	no	—
212	C₄²	C₂²⋊D₇	1	1	yes	characteristic
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	C₂×C₇	—	2	2	no	—
217	C₂×C₇	—	8	1	no	—
218	C₂×C₇	#525	1	1	yes	characteristic
219	C₂×C₇	C₂×#135	1	1	yes	characteristic
220	C₂×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
221	D₇	—	4	4	no	—
222	C₂³	—	2	1	no	—
223	C₂³	—	14	4	no	—
224	C₂³	—	14	2	no	—
225	C₂³	—	28	2	no	—
226	C₂³	—	28	2	no	—
227	C₂³	C₂×C₂²⋊D₇	1	1	yes	characteristic
228	Q₈	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	7	4	no	—
232	D₄	—	14	4	no	—
233	D₄	—	14	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	14	2	no	—
242	C₂×C₄	—	28	4	no	—
243	C₂×C₄	—	28	2	no	—
244	C₂×C₄	—	28	4	no	—
245	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
246	C₂×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
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	C₂³×D₇	1	1	yes	characteristic
251	C₈	—	14	2	no	—
252	C₈	—	14	2	no	—
253	C₇	#3105	1	1	yes	characteristic, complemented, Sylow
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	4	1	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	14	4	no	—
260	C₂²	—	28	8	no	—
261	C₂²	—	28	2	no	—
262	C₂²	—	28	2	no	—
263	C₂²	#6513	1	2	yes	—
264	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	28	2	no	—
273	C₄	#6512	1	2	yes	—
274	C₄	D₇×#41	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	—	8	1	no	—
277	C₂	—	28	4	no	—
278	C₂	C₂×#6512	1	1	yes	characteristic
279	C₂	C₂×#6513	1	1	yes	characteristic
280	C₂	#90519	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇
commutator, cosocle	C₂×C₄×C₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₇
Fitting	C₇×#529	C₂
socle	C₂²×C₇	C₂²×D₄
prime radical	#89808	C₂
hypercenter	#529	D₇
hypocenter	C₇	#3105
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ 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. C₂×C₄×D₂₈	C₂	semidirect
3. #89838	C₂	semidirect
4. #89829	C₂	semidirect
5. #89818	C₂	semidirect
6. #89816	C₂	semidirect
8. #89806	C₂	semidirect
9. C₄×#1143	C₂	semidirect
10. C₄×#1141	C₂	semidirect
11. #89697	C₂	semidirect
15. C₇×#125	C₂²	semidirect
16. C₇×#124	C₂²	semidirect
20. C₄×C₇×Q₈	C₂²	semidirect
22. C₄×C₇×D₄	C₂²	semidirect
33. #6409	C₂²	semidirect
34. #6406	C₂²	semidirect
39. #6382	C₂²	semidirect
40. #6381	C₂²	semidirect
41. C₄⋊(C₇⋊C₈)	C₂²	semidirect
42. C₄×C₇⋊C₈	C₂²	semidirect
83. #529	D₇	semidirect
130. #125	C₂×D₇	semidirect
131. #124	C₂×D₇	semidirect
135. C₄×Q₈	C₂×D₇	semidirect
141. C₄×D₄	C₂×D₇	semidirect
253. C₇	#3105	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
C₂×C₄×D₂₈	2
#89838	2
#89829	2
#89818	2
#89816	2
#89808	2
#89806	2
C₄×#1143	2
C₄×#1141	2
#89697	2
#3105	7