Subgroups — #13352834

Order 1792.

990 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11728219	C₂	1	3	yes	maximal
5	Dic₇×C₄⋊C₈	C₂	1	3	yes	maximal
6	C₈×C₄⋊Dic₇	C₂	1	3	yes	maximal
7	#11727504	C₂	1	3	yes	maximal
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈×Dic₇	C₄	1	2	yes	—
10	C₇×C₈×Q₈	C₄	1	8	yes	complemented
11	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	3	yes	—
13	C₂×C₈×Dic₇	C₂²	1	6	yes	—
14	Q₈×C₇⋊C₈	C₄	1	8	yes	complemented
15	C₄⋊C₄×Dic₇	C₄	1	6	yes	—
16	C₄×C₄⋊Dic₇	C₂²	1	3	yes	—
17	C₄²×Dic₇	C₂²	1	3	yes	—
18	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	3	yes	—
19	C₂×C₄×C₇⋊C₈	C₂²	1	3	yes	—
20	#89345	C₂²	1	3	yes	—
21	#89345	C₂²	1	3	yes	—
22	#89345	C₂²	1	1	yes	characteristic
23	#89345	C₂²	1	3	yes	—
24	#20569	—	7	1	no	maximal, Sylow
25	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	3	yes	—
28	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
29	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₇	C₂³	1	3	yes	—
32	Q₈×Dic₇	C₈	1	16	yes	complemented
33	C₂×C₄⋊Dic₇	C₂×C₄	1	6	yes	—
34	C₂×C₄×Dic₇	C₂×C₄	1	6	yes	—
35	C₂×C₄×Dic₇	C₂×C₄	1	6	yes	—
36	C₂×C₄×Dic₇	C₂³	1	1	yes	characteristic
37	C₂×C₄×Dic₇	C₂³	1	3	yes	—
38	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇⋊C₈	C₂³	1	3	yes	—
40	C₇×C₄⋊C₈	C₂×C₄	1	12	yes	—
41	C₄×C₇×C₈	C₂×C₄	1	12	yes	—
42	C₇⋊#95	C₂×C₄	1	6	yes	—
43	C₈×Dic₇	—	4	6	no	—
44	C₄⋊(C₇⋊C₈)	C₂×C₄	1	12	yes	—
45	C₄×C₇⋊C₈	C₂×C₄	1	12	yes	—
46	C₂×C₈×Q₈	—	7	1	no	—
47	C₂×C₈×Q₈	D₇	1	1	yes	characteristic
48	C₄²×Q₈	—	7	1	no	—
49	C₈×C₄⋊C₄	—	7	3	no	—
50	C₈×C₄⋊C₄	—	7	3	no	—
51	C₄×C₄⋊C₈	—	7	3	no	—
52	#1744	—	7	3	no	—
53	C₂×C₇×Q₈	C₄²	1	4	yes	—
54	C₂×C₇×Q₈	C₂×C₈	1	8	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
58	C₂²×Dic₇	C₂²×C₄	1	6	yes	—
59	C₂²×Dic₇	C₂²×C₄	1	2	yes	—
60	C₂×C₇×C₈	—	2	3	no	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	2	3	no	—
64	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	6	yes	—
66	C₇×C₄⋊C₄	C₄²	1	12	yes	—
67	C₄²×C₇	C₂²×C₄	1	6	yes	—
68	C₄²×C₇	C₂²×C₄	1	6	yes	—
69	C₄⋊Dic₇	C₂×C₈	1	24	yes	—
70	C₄×Dic₇	—	2	3	no	—
71	C₄×Dic₇	—	2	1	no	—
72	C₄×Dic₇	—	2	3	no	—
73	C₄×Dic₇	—	2	1	no	—
74	C₄×Dic₇	—	4	6	no	—
75	C₄×Dic₇	C₂×C₈	1	24	yes	—
76	C₂×C₇⋊C₈	—	2	3	no	—
77	C₂×C₇⋊C₈	—	2	1	no	—
78	C₂×C₇⋊C₈	—	2	1	no	—
79	C₂×C₇⋊C₈	—	2	3	no	—
80	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
81	C₂×C₇⋊C₈	C₂²×C₄	1	6	yes	—
82	C₂×C₄×Q₈	—	7	2	no	—
83	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
84	C₈×Q₈	—	7	8	no	—
85	C₈×Q₈	Dic₇	1	8	yes	complemented
86	C₂×C₄⋊C₈	—	7	3	no	—
87	C₂×C₄⋊C₈	C₂×D₇	1	3	yes	—
88	C₂×C₄×C₈	—	7	3	no	—
89	C₂×C₄×C₈	—	7	6	no	—
90	C₂×C₄×C₈	C₂×D₇	1	3	yes	—
91	C₄×C₄⋊C₄	—	7	6	no	—
92	C₄×C₄⋊C₄	—	7	3	no	—
93	C₄³	—	7	3	no	—
94	#336	—	7	1	no	—
95	#336	—	7	3	no	—
96	#336	—	7	3	no	—
97	#336	—	7	3	no	—
98	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
99	C₇×Q₈	C₄×C₈	1	16	yes	complemented
100	C₂×C₄×C₇	—	2	6	no	—
101	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
102	C₂×C₄×C₇	C₂×C₄²	1	6	yes	—
103	C₂×C₄×C₇	C₂²×C₈	1	12	yes	—
104	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
107	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
108	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
109	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×Dic₇	—	2	2	no	—
111	C₂×Dic₇	—	2	6	no	—
112	C₂×Dic₇	—	2	2	no	—
113	C₂×Dic₇	—	2	6	no	—
114	C₂×Dic₇	C₂²×C₈	1	4	yes	—
115	C₂×Dic₇	C₂²×C₈	1	12	yes	—
116	C₇×C₈	—	2	2	no	—
117	C₇×C₈	—	2	6	no	—
118	C₇⋊C₈	—	2	2	no	—
119	C₇⋊C₈	—	2	6	no	—
120	C₂²×Q₈	C₄×D₇	1	1	yes	characteristic
121	C₂²×C₈	—	7	1	no	—
122	C₂²×C₈	—	7	3	no	—
123	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₇	1	3	yes	—
125	C₄×Q₈	—	7	16	no	—
126	C₄×Q₈	C₄×D₇	1	4	yes	—
127	C₄×Q₈	C₂×Dic₇	1	4	yes	—
128	C₂×C₄⋊C₄	—	7	6	no	—
129	C₂×C₄⋊C₄	C₄×D₇	1	3	yes	—
130	C₂×C₄²	—	7	6	no	—
131	C₂×C₄²	—	7	1	no	—
132	C₂×C₄²	—	7	3	no	—
133	C₂×C₄²	—	7	6	no	—
134	C₂×C₄²	C₂²×D₇	1	3	yes	—
135	C₄⋊C₈	—	7	12	no	—
136	C₄⋊C₈	C₂×Dic₇	1	12	yes	—
137	C₄×C₈	—	7	12	no	—
138	C₄×C₈	—	28	6	no	—
139	C₄×C₈	C₂×Dic₇	1	12	yes	—
140	#95	—	7	6	no	—
141	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
142	C₂²×C₇	#519	1	1	yes	characteristic
143	C₂²×C₇	#520	1	1	yes	characteristic
144	C₂²×C₇	C₂³×C₈	1	2	yes	—
145	C₂²×C₇	#568	1	2	yes	—
146	C₄×C₇	—	2	6	no	—
147	C₄×C₇	C₂×C₄×C₈	1	12	yes	—
148	C₄×C₇	#519	1	1	yes	characteristic
149	C₄×C₇	#519	1	1	yes	characteristic
150	C₄×C₇	#520	1	1	yes	characteristic
151	C₄×C₇	#520	1	1	yes	characteristic
152	Dic₇	—	2	4	no	—
153	Dic₇	—	2	12	no	—
154	C₂×Q₈	C₈×D₇	1	8	yes	—
155	C₂×Q₈	C₄×Dic₇	1	4	yes	—
156	C₂²×C₄	—	7	6	no	—
157	C₂²×C₄	—	7	2	no	—
158	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
159	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
160	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	3	no	—
164	C₂×C₈	—	2	3	no	—
165	C₂×C₈	—	7	2	no	—
166	C₂×C₈	—	7	6	no	—
167	C₂×C₈	—	14	1	no	—
168	C₂×C₈	—	14	3	no	—
169	C₂×C₈	—	14	3	no	—
170	C₂×C₈	—	14	1	no	—
171	C₂×C₈	C₂²×Dic₇	1	2	yes	—
172	C₂×C₈	C₂²×Dic₇	1	6	yes	—
173	C₄⋊C₄	—	7	24	no	—
174	C₄⋊C₄	C₄×Dic₇	1	12	yes	—
175	C₄²	—	7	24	no	—
176	C₄²	—	14	1	no	—
177	C₄²	—	14	3	no	—
178	C₄²	—	14	3	no	—
179	C₄²	—	14	1	no	—
180	C₄²	—	28	6	no	—
181	C₄²	C₂×C₄×D₇	1	6	yes	—
182	C₄²	C₂²×Dic₇	1	6	yes	—
183	C₂×C₇	#2276	1	1	yes	characteristic
184	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
185	C₂×C₇	#2875	1	1	yes	characteristic
186	C₂×C₇	#2982	1	2	yes	—
187	C₂×C₇	#2988	1	2	yes	—
188	C₂³	C₂²×C₄×D₇	1	1	yes	characteristic
189	Q₈	C₈×Dic₇	1	16	yes	complemented
190	C₂×C₄	—	2	6	no	—
191	C₂×C₄	—	7	4	no	—
192	C₂×C₄	—	7	12	no	—
193	C₂×C₄	—	14	2	no	—
194	C₂×C₄	—	14	6	no	—
195	C₂×C₄	—	14	2	no	—
196	C₂×C₄	—	14	6	no	—
197	C₂×C₄	C₂×C₈×D₇	1	12	yes	—
198	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
199	C₂×C₄	C₂×C₄×Dic₇	1	6	yes	—
200	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
201	C₂×C₄	#6551	1	1	yes	characteristic
202	C₂×C₄	#6551	1	1	yes	characteristic
203	C₂×C₄	#6553	1	1	yes	characteristic
204	C₂×C₄	#6554	1	1	yes	characteristic
205	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
206	C₈	—	2	2	no	—
207	C₈	—	2	6	no	—
208	C₈	—	14	2	no	—
209	C₈	—	14	6	no	—
210	C₇	#20569	1	1	yes	characteristic, complemented, Sylow
211	C₂²	#90263	1	1	yes	characteristic
212	C₂²	#90265	1	1	yes	characteristic
213	C₂²	C₂²×C₈×D₇	1	2	yes	—
214	C₂²	#90441	1	2	yes	—
215	C₂²	C₂²×C₄×Dic₇	1	1	yes	characteristic
216	C₄	—	2	6	no	—
217	C₄	—	14	12	no	—
218	C₄	—	14	4	no	—
219	C₄	C₂×C₈×Dic₇	1	12	yes	—
220	C₄	#90263	1	1	yes	characteristic
221	C₄	#90265	1	1	yes	characteristic
222	C₄	#90503	1	1	yes	characteristic
223	C₄	#90518	1	1	yes	characteristic
224	C₂	#11737351	1	2	yes	—
225	C₂	#11737365	1	2	yes	—
226	C₂	#11739895	1	1	yes	characteristic
227	C₂	C₂²×C₈×Dic₇	1	1	yes	characteristic
228	C₂	#11741227	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇
commutator	C₂×C₇	C₂²×C₄×C₈
Fitting	C₂×C₇×C₈×Q₈	C₂
socle, prime radical	C₂³×C₇	C₂³×C₄
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇
hypocenter	C₇	#20569
LCS	G ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. C₇×C₈×Q₈	C₄	semidirect
14. Q₈×C₇⋊C₈	C₄	semidirect
32. Q₈×Dic₇	C₈	semidirect
85. C₈×Q₈	Dic₇	semidirect
99. C₇×Q₈	C₄×C₈	semidirect
189. Q₈	C₈×Dic₇	semidirect
210. C₇	#20569	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×C₈×Q₈	2
C₄×Q₈×Dic₇	2
C₂×Q₈×C₇⋊C₈	2
#11728219	2
Dic₇×C₄⋊C₈	2
C₈×C₄⋊Dic₇	2
#11727504	2
#20569	7