Subgroups — #11736526

Order 896.

484 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6470	C₂	1	1	yes	maximal, characteristic
2	C₂×#6464	C₂	1	1	yes	maximal, characteristic
3	#90213	C₂	1	1	yes	maximal, characteristic
4	C₇×#424	C₂	1	1	yes	maximal, characteristic, complemented
5	#89890	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#89771	C₂	1	1	yes	maximal, characteristic, complemented
8	#89611	C₂	1	2	yes	maximal, complemented
9	#89611	C₂	1	2	yes	maximal, complemented
10	#89609	C₂	1	2	yes	maximal, complemented
11	D₇×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
12	C₂×#1157	C₂²	1	1	yes	characteristic
13	C₂×C₇×M₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₇×#117	C₂²	1	1	yes	characteristic
16	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×#1136	C₂²	1	1	yes	characteristic
18	C₂²×C₇⋊C₈	C₂²	1	1	yes	characteristic
19	#6470	—	2	4	no	—
20	#6464	—	2	4	no	—
21	C₂×#1131	C₂²	1	2	yes	—
22	C₂×#1131	C₂²	1	2	yes	—
23	C₂×C₈×D₇	C₂²	1	2	yes	—
24	C₂×C₈×D₇	C₂²	1	2	yes	—
25	#6456	C₄	1	1	yes	characteristic
26	#6456	C₄	1	1	yes	characteristic
27	#6450	C₄	1	1	yes	characteristic
28	#6450	C₄	1	1	yes	characteristic
29	(C₂²⋊D₇)⋊C₄	C₄	1	2	yes	—
30	(C₂²⋊D₇)⋊C₄	C₄	1	2	yes	—
31	#6435	C₂²	1	2	yes	—
32	C₄²×D₇	C₂²	1	2	yes	—
33	C₇×C₄⋊C₈	C₂²	1	2	yes	—
34	C₇×C₄⋊C₈	C₂²	1	2	yes	—
35	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
36	Dic₇⋊C₈	C₂²	1	2	yes	—
37	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
38	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
39	Dic₇⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₇⋊C₈	C₂²	1	2	yes	—
41	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
42	C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
43	#2906	—	7	1	no	maximal, Sylow
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂×C₂²⋊D₇	C₂×C₄	1	2	yes	—
46	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₇	C₂×C₄	1	1	yes	characteristic
48	#1157	D₄	1	4	yes	—
49	#1157	Q₈	1	4	yes	—
50	C₂×D₂₈	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
52	C₂×C₄×D₇	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₇	C₂³	1	2	yes	—
54	C₂×Dic₁₄	C₂×C₄	1	1	yes	characteristic
55	C₇×M₁₆	—	2	2	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂³	1	2	yes	—
59	C₂×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
61	C₇×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
62	C₇×C₂²⋊C₄	C₂×C₄	1	2	yes	—
63	C₄²×C₇	C₂³	1	2	yes	—
64	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
65	(C₂×D₇)⋊C₄	C₂×C₄	1	2	yes	—
66	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
67	Dic₇⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₇	C₂×C₄	1	2	yes	—
69	C₄×Dic₇	C₂×C₄	1	2	yes	—
70	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
71	C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
72	C₄×Dic₇	C₂³	1	1	yes	characteristic
73	C₄×Dic₇	C₂³	1	1	yes	characteristic
74	#1136	—	2	2	no	—
75	C₂×C₇⋊C₈	—	2	2	no	—
76	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₂×C₇⋊C₈	C₂³	1	2	yes	—
78	C₂×C₇⋊C₈	C₂³	1	1	yes	characteristic
79	#1131	—	2	4	no	—
80	#1131	—	2	4	no	—
81	C₈×D₇	—	2	4	no	—
82	C₈×D₇	—	2	4	no	—
83	C₂×#131	—	7	1	no	—
84	C₂×#131	—	7	1	no	—
85	C₄×#41	—	7	1	no	—
86	#424	—	7	1	no	—
87	#424	—	7	2	no	—
88	#424	—	7	2	no	—
89	#424	D₇	1	1	yes	characteristic, complemented
90	#423	—	7	1	no	—
91	#423	—	7	2	no	—
92	C₂×C₄⋊C₈	—	7	1	no	—
93	C₂×C₄⋊C₈	—	7	2	no	—
94	C₂³×C₇	C₂²×C₄	1	1	yes	characteristic
95	C₂²×D₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
97	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
98	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
101	C₂×C₄×C₇	C₂⁴	1	1	yes	characteristic
102	C₂²⋊D₇	C₄⋊C₄	1	8	yes	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	4	1	no	—
105	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
108	C₂×Dic₇	C₂²×C₄	1	2	yes	—
109	C₂×Dic₇	C₂²×C₄	1	1	yes	characteristic
110	D₂₈	C₄⋊C₄	1	4	yes	—
111	C₄×D₇	—	2	4	no	—
112	C₄×D₇	—	4	2	no	—
113	C₄×D₇	C₂×D₄	1	4	yes	—
114	C₄×D₇	C₂×Q₈	1	4	yes	—
115	Dic₁₄	C₄⋊C₄	1	4	yes	—
116	C₇×C₈	—	2	1	no	—
117	C₇×C₈	—	2	2	no	—
118	C₇×C₈	—	2	1	no	—
119	C₇⋊C₈	—	2	1	no	—
120	C₇⋊C₈	—	2	2	no	—
121	C₇⋊C₈	—	2	1	no	—
122	C₂×#41	—	7	1	no	—
123	#131	—	14	4	no	—
124	#131	—	14	4	no	—
125	C₂×M₁₆	—	7	1	no	—
126	C₂×M₁₆	—	7	2	no	—
127	C₂×M₁₆	—	7	2	no	—
128	C₂×M₁₆	C₂×D₇	1	1	yes	characteristic
129	C₂²×C₈	—	7	1	no	—
130	C₂²×C₈	—	7	2	no	—
131	C₂²×C₈	—	7	2	no	—
132	C₂²×C₈	C₂×D₇	1	1	yes	characteristic
133	C₄×Q₈	—	7	1	no	—
134	C₄×Q₈	—	7	1	no	—
135	C₄×D₄	—	7	1	no	—
136	C₄×D₄	—	7	1	no	—
137	C₄×D₄	—	7	2	no	—
138	C₄×D₄	—	7	2	no	—
139	#117	—	7	2	no	—
140	#117	C₂×D₇	1	1	yes	characteristic
141	C₂×C₄²	—	7	1	no	—
142	C₂×C₄²	—	7	2	no	—
143	C₄⋊C₈	—	7	1	no	—
144	C₄⋊C₈	—	7	2	no	—
145	C₄⋊C₈	—	7	1	no	—
146	C₄⋊C₈	—	7	1	no	—
147	C₄⋊C₈	—	7	1	no	—
148	C₄⋊C₈	—	7	2	no	—
149	C₄⋊C₈	—	7	2	no	—
150	C₄⋊C₈	—	7	2	no	—
151	C₄⋊C₈	C₂×D₇	1	2	yes	—
152	C₄⋊C₈	C₂×D₇	1	2	yes	—
153	C₂²×C₇	—	2	1	no	—
154	C₂²×C₇	—	2	1	no	—
155	C₂²×C₇	C₂×C₄⋊C₄	1	2	yes	—
156	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
157	C₂×D₇	—	2	2	no	—
158	C₂×D₇	—	2	2	no	—
159	C₂×D₇	C₂×C₄⋊C₄	1	4	yes	—
160	C₄×C₇	—	2	2	no	—
161	C₄×C₇	—	2	2	no	—
162	C₄×C₇	C₂×C₄⋊C₄	1	2	yes	—
163	C₄×C₇	C₂²×D₄	1	1	yes	characteristic
164	C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
165	Dic₇	—	2	1	no	—
166	Dic₇	—	2	1	no	—
167	Dic₇	—	2	1	no	—
168	Dic₇	C₂×C₄⋊C₄	1	4	yes	—
169	Dic₇	#131	1	2	yes	complemented
170	#41	—	7	4	no	—
171	#41	—	7	4	no	—
172	C₂×Q₈	—	7	1	no	—
173	C₂×D₄	—	7	2	no	—
174	C₂×D₄	—	7	1	no	—
175	C₂²×C₄	—	7	2	no	—
176	C₂²×C₄	—	7	1	no	—
177	C₂²×C₄	—	7	1	no	—
178	C₂²×C₄	—	7	2	no	—
179	C₂²×C₄	—	7	2	no	—
180	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
181	M₁₆	—	2	2	no	—
182	M₁₆	—	14	4	no	—
183	M₁₆	—	14	4	no	—
184	M₁₆	—	14	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	7	1	no	—
187	C₂×C₈	—	7	1	no	—
188	C₂×C₈	—	7	2	no	—
189	C₂×C₈	—	14	4	no	—
190	C₂×C₈	—	14	4	no	—
191	C₂×C₈	—	14	2	no	—
192	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
193	C₂×C₈	C₂²×D₇	1	2	yes	—
194	C₂×C₈	C₂²×D₇	1	1	yes	characteristic
195	C₄⋊C₄	—	7	2	no	—
196	C₄⋊C₄	—	7	2	no	—
197	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
198	C₄⋊C₄	C₄×D₇	1	1	yes	characteristic
199	C₂²⋊C₄	—	7	2	no	—
200	C₂²⋊C₄	—	7	2	no	—
201	C₂²⋊C₄	C₄×D₇	1	2	yes	—
202	C₄²	—	7	2	no	—
203	C₄²	—	7	1	no	—
204	C₄²	—	7	2	no	—
205	C₄²	—	7	1	no	—
206	C₄²	—	7	1	no	—
207	C₄²	—	7	1	no	—
208	C₄²	C₂²×D₇	1	2	yes	—
209	C₂×C₇	—	2	2	no	—
210	C₂×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
211	C₂×C₇	C₂×#131	1	1	yes	characteristic
212	C₂×C₇	#568	1	1	yes	characteristic
213	D₇	—	2	4	no	—
214	C₂³	—	7	2	no	—
215	C₂³	C₂×C₄×D₇	1	1	yes	characteristic
216	Q₈	—	7	4	no	—
217	D₄	—	7	8	no	—
218	D₄	—	7	4	no	—
219	C₂×C₄	—	7	1	no	—
220	C₂×C₄	—	7	2	no	—
221	C₂×C₄	—	7	4	no	—
222	C₂×C₄	—	7	4	no	—
223	C₂×C₄	—	7	1	no	—
224	C₂×C₄	—	7	1	no	—
225	C₂×C₄	—	7	1	no	—
226	C₂×C₄	—	14	4	no	—
227	C₂×C₄	—	14	2	no	—
228	C₂×C₄	—	28	2	no	—
229	C₂×C₄	—	28	1	no	—
230	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
231	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
232	C₂×C₄	C₂×C₄×D₇	1	1	yes	characteristic
233	C₂×C₄	D₄×D₇	1	2	yes	—
234	C₂×C₄	Q₈×D₇	1	2	yes	—
235	C₂×C₄	C₂³×D₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	—	2	2	no	—
239	C₈	—	14	1	no	—
240	C₈	—	14	2	no	—
241	C₈	—	14	1	no	—
242	C₇	#2906	1	1	yes	characteristic, complemented, Sylow
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	7	4	no	—
246	C₂²	—	14	2	no	—
247	C₂²	—	14	2	no	—
248	C₂²	D₇×C₄⋊C₄	1	2	yes	—
249	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	7	2	no	—
253	C₄	—	7	4	no	—
254	C₄	—	14	1	no	—
255	C₄	—	14	1	no	—
256	C₄	—	14	1	no	—
257	C₄	D₇×C₄⋊C₄	1	2	yes	—
258	C₄	C₂×D₄×D₇	1	1	yes	characteristic
259	C₄	C₂×Q₈×D₇	1	1	yes	characteristic
260	C₂	—	2	2	no	—
261	C₂	—	14	4	no	—
262	C₂	C₂×D₇×C₄⋊C₄	1	1	yes	characteristic
263	C₂	D₇×#131	1	1	yes	characteristic
264	C₂	#90441	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇
commutator, socle	C₂²×C₇	C₂³×C₄
Fitting	C₇×#424	C₂
prime radical	C₂×#1157	C₂²
cosocle	C₂×C₄×C₇	C₂⁴
hypercenter	#424	D₇
hypocenter	C₇	#2906
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ #424

Complements

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

N	H ≅ G/N	type
4. C₇×#424	C₂	semidirect
5. #89890	C₂	semidirect
6. C₂×Dic₇⋊C₈	C₂	semidirect
7. #89771	C₂	semidirect
8. #89611	C₂	semidirect
9. #89611	C₂	semidirect
10. #89609	C₂	semidirect
11. D₇×C₄⋊C₈	C₂	semidirect
89. #424	D₇	semidirect
169. Dic₇	#131	semidirect
242. C₇	#2906	semidirect

Maximal subgroups

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

subgroup	index
C₂×#6470	2
C₂×#6464	2
#90213	2
C₇×#424	2
#89890	2
C₂×Dic₇⋊C₈	2
#89771	2
#89611	2
#89609	2
D₇×C₄⋊C₈	2
#2906	7