Subgroups — #11740687

Order 896.

556 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×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#6507	C₂	1	2	yes	maximal
3	C₂×#6451	C₂	1	1	yes	maximal, characteristic
4	#90001	C₂	1	2	yes	maximal
5	#89751	C₂	1	2	yes	maximal
6	#89748	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₇	C₂	1	1	yes	maximal, characteristic
8	#89745	C₂	1	1	yes	maximal, characteristic
9	#89744	C₂	1	2	yes	maximal
10	C₄×Dic₇⋊C₄	C₂	1	1	yes	maximal, characteristic
11	#89700	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₇×Q₈	C₂²	1	2	yes	—
13	C₂²×Dic₁₄	C₂²	1	2	yes	—
14	C₇×C₄⋊Q₈	—	2	4	no	—
15	C₂×C₇×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
19	#6507	—	2	8	no	—
20	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
22	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
24	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
25	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
26	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
27	C₂×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₇⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₇	C₂²	1	2	yes	—
31	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
32	#6451	—	2	4	no	—
33	C₇⋊#95	C₂²	1	2	yes	—
34	C₇⋊#95	C₂²	1	1	yes	characteristic
35	C₇⋊#95	C₂²	1	1	yes	characteristic
36	C₇⋊#95	C₂²	1	2	yes	—
37	C₇⋊#95	C₂²	1	2	yes	—
38	#2551	—	7	1	no	maximal, Sylow
39	C₂×C₇×Q₈	—	2	4	no	—
40	C₂×C₇×Q₈	—	2	4	no	—
41	C₂×C₇×Q₈	—	4	2	no	—
42	C₂²×C₄×C₇	C₂³	1	2	yes	—
43	C₂²×C₄×C₇	C₂³	1	2	yes	—
44	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
47	C₂²×Dic₇	C₂³	1	2	yes	—
48	C₂²×Dic₇	C₂³	1	1	yes	characteristic
49	C₂²×Dic₇	C₂³	1	1	yes	characteristic
50	C₂²×Dic₇	C₂³	1	1	yes	characteristic
51	C₂²×Dic₇	C₂³	1	2	yes	—
52	C₂²×Dic₇	C₂³	1	1	yes	characteristic
53	C₂×Dic₁₄	—	2	4	no	—
54	C₂×Dic₁₄	—	2	4	no	—
55	C₂×Dic₁₄	—	4	2	no	—
56	C₇×C₄⋊C₄	—	2	4	no	—
57	C₇×C₄⋊C₄	—	2	2	no	—
58	C₇×C₄⋊C₄	D₄	1	4	yes	—
59	C₄²×C₇	—	2	2	no	—
60	C₄⋊Dic₇	—	2	2	no	—
61	C₄⋊Dic₇	Q₈	1	4	yes	complemented
62	Dic₇⋊C₄	—	2	4	no	—
63	Dic₇⋊C₄	—	2	4	no	—
64	Dic₇⋊C₄	—	2	4	no	—
65	Dic₇⋊C₄	—	2	4	no	—
66	Dic₇⋊C₄	—	4	2	no	—
67	Dic₇⋊C₄	D₄	1	4	yes	—
68	Dic₇⋊C₄	Q₈	1	4	yes	complemented
69	C₄×Dic₇	—	2	2	no	—
70	C₄×Dic₇	—	2	4	no	—
71	C₄×Dic₇	Q₈	1	4	yes	complemented
72	C₂×C₄⋊Q₈	—	7	2	no	—
73	C₂×C₄⋊Q₈	—	7	1	no	—
74	C₂×C₄⋊Q₈	D₇	1	1	yes	characteristic
75	#395	—	7	2	no	—
76	#395	—	7	2	no	—
77	#389	—	7	2	no	—
78	#389	—	7	1	no	—
79	#389	—	7	1	no	—
80	#384	—	7	1	no	—
81	C₄×C₄⋊C₄	—	7	1	no	—
82	C₄×C₄⋊C₄	—	7	1	no	—
83	C₂³×C₇	C₂⁴	1	1	yes	characteristic
84	C₇×Q₈	—	4	8	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	1	no	—
87	C₂×C₄×C₇	—	2	1	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	1	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	2	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₇	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
97	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
98	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
99	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
100	C₂×Dic₇	—	2	2	no	—
101	C₂×Dic₇	—	2	1	no	—
102	C₂×Dic₇	—	2	1	no	—
103	C₂×Dic₇	—	2	1	no	—
104	C₂×Dic₇	—	2	1	no	—
105	C₂×Dic₇	—	2	1	no	—
106	C₂×Dic₇	—	2	1	no	—
107	C₂×Dic₇	—	2	2	no	—
108	C₂×Dic₇	—	2	2	no	—
109	C₂×Dic₇	—	2	2	no	—
110	C₂×Dic₇	—	2	2	no	—
111	C₂×Dic₇	C₂×D₄	1	2	yes	—
112	C₂×Dic₇	C₂×D₄	1	2	yes	—
113	C₂×Dic₇	C₂×Q₈	1	2	yes	—
114	C₂×Dic₇	C₂×Q₈	1	2	yes	—
115	C₂×Dic₇	C₂×Q₈	1	4	yes	—
116	C₂×Dic₇	C₂×Q₈	1	2	yes	—
117	C₂×Dic₇	C₂×Q₈	1	2	yes	—
118	Dic₁₄	—	4	8	no	—
119	C₂²×Q₈	—	7	2	no	—
120	C₂²×Q₈	C₂×D₇	1	2	yes	—
121	C₄⋊Q₈	—	2	4	no	—
122	C₄⋊Q₈	—	14	8	no	—
123	C₄⋊Q₈	—	14	4	no	—
124	C₂×C₄⋊C₄	—	7	1	no	—
125	C₂×C₄⋊C₄	—	7	1	no	—
126	C₂×C₄⋊C₄	—	7	1	no	—
127	C₂×C₄⋊C₄	—	7	2	no	—
128	C₂×C₄⋊C₄	—	7	2	no	—
129	C₂×C₄⋊C₄	—	7	2	no	—
130	C₂×C₄⋊C₄	—	7	2	no	—
131	C₂×C₄⋊C₄	—	7	1	no	—
132	C₂×C₄⋊C₄	—	7	2	no	—
133	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
134	C₂×C₄⋊C₄	C₂×D₇	1	2	yes	—
135	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
136	C₂×C₄²	—	7	1	no	—
137	C₂×C₄²	—	7	2	no	—
138	C₂×C₄²	—	7	1	no	—
139	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
140	#95	—	7	1	no	—
141	#95	—	7	2	no	—
142	#95	—	7	1	no	—
143	#95	—	7	2	no	—
144	#95	—	7	2	no	—
145	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
146	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
147	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
149	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
150	C₂²×C₇	H(5,𝔽₂)	1	1	yes	characteristic
151	C₂²×C₇	2₋¹⁺⁴	1	1	yes	characteristic
152	C₄×C₇	—	2	4	no	—
153	C₄×C₇	—	2	2	no	—
154	C₄×C₇	—	2	2	no	—
155	C₄×C₇	—	4	2	no	—
156	C₄×C₇	C₄⋊Q₈	1	4	yes	complemented
157	Dic₇	—	2	2	no	—
158	Dic₇	—	2	2	no	—
159	Dic₇	—	2	4	no	—
160	Dic₇	—	2	2	no	—
161	Dic₇	—	4	2	no	—
162	Dic₇	C₄⋊Q₈	1	4	yes	complemented
163	C₂×Q₈	—	2	4	no	—
164	C₂×Q₈	—	2	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂×Q₈	—	14	4	no	—
167	C₂×Q₈	—	14	4	no	—
168	C₂×Q₈	—	28	2	no	—
169	C₂²×C₄	—	7	1	no	—
170	C₂²×C₄	—	7	1	no	—
171	C₂²×C₄	—	7	2	no	—
172	C₂²×C₄	—	7	1	no	—
173	C₂²×C₄	—	7	1	no	—
174	C₂²×C₄	—	7	2	no	—
175	C₂²×C₄	C₂²×D₇	1	2	yes	—
176	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
177	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
178	C₂²×C₄	C₂²×D₇	1	2	yes	—
179	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	—	7	4	no	—
183	C₄⋊C₄	—	7	4	no	—
184	C₄⋊C₄	—	7	4	no	—
185	C₄⋊C₄	—	14	4	no	—
186	C₄⋊C₄	—	14	4	no	—
187	C₄⋊C₄	—	14	4	no	—
188	C₄⋊C₄	—	14	4	no	—
189	C₄⋊C₄	—	14	2	no	—
190	C₄⋊C₄	—	28	2	no	—
191	C₄⋊C₄	C₂²⋊D₇	1	4	yes	—
192	C₄²	—	2	2	no	—
193	C₄²	—	7	4	no	—
194	C₄²	—	14	4	no	—
195	C₄²	—	14	2	no	—
196	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
197	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
198	C₂×C₇	#544	1	1	yes	characteristic
199	C₂×C₇	D₄²	1	1	yes	characteristic
200	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
201	C₂×C₇	D₄×Q₈	1	1	yes	characteristic
202	C₂×C₇	Q₈²	1	1	yes	characteristic
203	C₂³	C₂³×D₇	1	1	yes	characteristic
204	Q₈	—	4	8	no	—
205	Q₈	—	28	8	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	1	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	7	2	no	—
216	C₂×C₄	—	7	2	no	—
217	C₂×C₄	—	7	2	no	—
218	C₂×C₄	—	7	2	no	—
219	C₂×C₄	—	7	2	no	—
220	C₂×C₄	—	7	4	no	—
221	C₂×C₄	—	7	2	no	—
222	C₂×C₄	—	14	2	no	—
223	C₂×C₄	—	14	1	no	—
224	C₂×C₄	—	14	1	no	—
225	C₂×C₄	—	14	2	no	—
226	C₂×C₄	—	14	1	no	—
227	C₂×C₄	—	14	1	no	—
228	C₂×C₄	—	14	1	no	—
229	C₂×C₄	—	14	1	no	—
230	C₂×C₄	—	14	2	no	—
231	C₂×C₄	—	14	2	no	—
232	C₂×C₄	—	14	2	no	—
233	C₂×C₄	D₄×D₇	1	2	yes	—
234	C₂×C₄	Q₈×D₇	1	2	yes	—
235	C₂×C₄	Q₈×D₇	1	2	yes	—
236	C₂×C₄	Q₈×D₇	1	2	yes	—
237	C₂×C₄	C₂×C₂²⋊D₇	1	4	yes	—
238	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
239	C₇	#2551	1	1	yes	characteristic, complemented, Sylow
240	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
241	C₂²	#6548	1	1	yes	characteristic
242	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
243	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
244	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
245	C₂²	#6551	1	1	yes	characteristic
246	C₂²	C₂²×C₂²⋊D₇	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	4	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	2	no	—
251	C₄	—	7	4	no	—
252	C₄	—	14	4	no	—
253	C₄	—	14	2	no	—
254	C₄	—	14	2	no	—
255	C₄	—	14	2	no	—
256	C₄	—	28	2	no	—
257	C₄	#6507	1	4	yes	complemented
258	C₂	#90413	1	1	yes	characteristic
259	C₂	D₇×C₄⋊Q₈	1	1	yes	characteristic
260	C₂	#90418	1	1	yes	characteristic
261	C₂	#90420	1	1	yes	characteristic
262	C₂	D₄×C₂²⋊D₇	1	1	yes	characteristic
263	C₂	C₂×#6507	1	1	yes	characteristic
264	C₂	Q₈×C₂²⋊D₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇
commutator, socle, prime radical, cosocle	C₂³×C₇	C₂⁴
Fitting	C₂×C₇×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₇
hypocenter	C₇	#2551
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
61. C₄⋊Dic₇	Q₈	semidirect
68. Dic₇⋊C₄	Q₈	semidirect
71. C₄×Dic₇	Q₈	semidirect
156. C₄×C₇	C₄⋊Q₈	semidirect
162. Dic₇	C₄⋊Q₈	semidirect
239. C₇	#2551	semidirect
257. C₄	#6507	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₄⋊Q₈	2
C₂×#6507	2
C₂×#6451	2
#90001	2
#89751	2
#89748	2
C₄⋊C₄×Dic₇	2
#89745	2
#89744	2
C₄×Dic₇⋊C₄	2
#89700	2
#2551	7