Subgroups — #13506369

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#500	C₂	1	1	yes	maximal, characteristic
2	#11740698	C₂	1	2	yes	maximal
3	C₂×#89850	C₂	1	2	yes	maximal
4	C₂×#89454	C₂	1	1	yes	maximal, characteristic
5	#11728360	C₂	1	8	yes	maximal
6	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×#6507	—	2	4	no	—
9	C₂×#6432	C₂²	1	2	yes	—
10	C₇×#500	—	2	4	no	—
11	C₂×C₇×#107	C₂²	1	4	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6392	C₂²	1	4	yes	—
14	C₂×C₈×Dic₇	C₂²	1	4	yes	—
15	#89850	—	2	8	no	—
16	#89748	—	2	4	no	—
17	#89748	C₂²	1	8	yes	—
18	C₂×#6381	C₂²	1	8	yes	—
19	C₄²×Dic₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₇⋊C₈	C₂²	1	1	yes	characteristic
21	#89454	—	2	4	no	—
22	#33312	—	7	1	no	maximal, Sylow
23	C₂²×C₇×Q₈	—	2	2	no	—
24	C₂²×Dic₁₄	—	2	2	no	—
25	C₂²×C₇×C₈	C₂³	1	2	yes	—
26	C₇×C₄⋊Q₈	—	2	4	no	—
27	C₇×C₄⋊Q₈	—	2	4	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
31	#6507	—	4	16	no	—
32	C₂×C₄⋊Dic₇	—	2	2	no	—
33	C₂×C₄⋊Dic₇	C₂³	1	4	yes	—
34	C₂×Dic₇⋊C₄	—	2	16	no	—
35	C₂×C₄×Dic₇	D₄	1	2	yes	—
36	C₂×C₄×Dic₇	D₄	1	2	yes	—
37	C₂×C₄×Dic₇	C₂³	1	2	yes	—
38	C₂²×C₇⋊C₈	C₂³	1	2	yes	—
39	#6432	—	2	4	no	—
40	#6432	—	2	4	no	—
41	C₇×#107	—	2	8	no	—
42	C₄×C₇×C₈	D₄	1	4	yes	—
43	#6392	—	2	8	no	—
44	C₈×Dic₇	Q₈	1	16	yes	complemented
45	#6381	—	2	16	no	—
46	C₄×C₇⋊C₈	D₄	1	4	yes	—
47	C₂×#500	—	7	1	no	—
48	C₂×#500	—	7	2	no	—
49	C₂×#500	D₇	1	1	yes	characteristic
50	#2868	—	7	2	no	—
51	#1845	—	7	8	no	—
52	C₄²×C₈	—	7	1	no	—
53	C₂×C₇×Q₈	—	4	4	no	—
54	C₂×C₇×Q₈	—	4	8	no	—
55	C₂²×C₄×C₇	—	2	4	no	—
56	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₂²×Dic₇	—	2	4	no	—
60	C₂²×Dic₇	C₂×D₄	1	4	yes	—
61	C₂×Dic₁₄	—	4	4	no	—
62	C₂×Dic₁₄	—	4	8	no	—
63	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
64	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
65	C₇×C₄⋊C₄	—	2	4	no	—
66	C₇×C₄⋊C₄	—	2	4	no	—
67	C₇×C₄⋊C₄	—	4	4	no	—
68	C₄²×C₇	C₂×D₄	1	2	yes	—
69	C₄²×C₇	C₂×D₄	1	2	yes	—
70	C₄⋊Dic₇	—	2	4	no	—
71	C₄⋊Dic₇	—	2	4	no	—
72	C₄⋊Dic₇	—	4	4	no	—
73	Dic₇⋊C₄	—	4	32	no	—
74	C₄×Dic₇	D₈	1	8	yes	—
75	C₄×Dic₇	Q₁₆	1	8	yes	—
76	C₄×Dic₇	C₂×Q₈	1	8	yes	—
77	C₂×C₇⋊C₈	C₂×D₄	1	4	yes	—
78	C₂×C₇⋊C₈	C₂×Q₈	1	8	yes	—
79	C₂×C₄⋊Q₈	—	7	2	no	—
80	C₂×C₄⋊Q₈	—	14	4	no	—
81	C₂×C₄⋊Q₈	C₂×D₇	1	2	yes	—
82	#500	—	2	4	no	—
83	#500	—	14	8	no	—
84	#500	—	14	4	no	—
85	C₂×#107	—	7	4	no	—
86	C₂×#107	—	7	8	no	—
87	C₂×#107	C₂×D₇	1	4	yes	—
88	C₂×C₄×C₈	—	7	4	no	—
89	C₂×C₄×C₈	—	7	1	no	—
90	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
91	#384	—	7	8	no	—
92	#384	—	14	4	no	—
93	C₄³	—	7	1	no	—
94	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
95	C₇×Q₈	—	8	8	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	8	no	—
98	C₂×C₄×C₇	#127	1	2	yes	—
99	C₂×C₄×C₇	#127	1	2	yes	—
100	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₇	C₂×Q₁₆	1	4	yes	—
102	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂²×Q₈	1	4	yes	—
105	C₂×Dic₇	—	4	4	no	—
106	C₂×Dic₇	—	4	8	no	—
107	C₂×Dic₇	C₄⋊Q₈	1	8	yes	—
108	C₂×Dic₇	C₂×D₈	1	8	yes	—
109	C₂×Dic₇	C₂×Q₁₆	1	8	yes	—
110	Dic₁₄	—	8	8	no	—
111	C₇×C₈	C₄⋊Q₈	1	8	yes	complemented
112	C₇⋊C₈	C₄⋊Q₈	1	8	yes	complemented
113	C₂²×Q₈	—	2	2	no	—
114	C₂²×Q₈	—	14	2	no	—
115	C₂²×C₈	—	7	2	no	—
116	C₂²×C₈	C₂²×D₇	1	2	yes	—
117	C₄⋊Q₈	—	2	4	no	—
118	C₄⋊Q₈	—	2	4	no	—
119	C₄⋊Q₈	—	14	4	no	—
120	C₄⋊Q₈	—	14	4	no	—
121	C₄⋊Q₈	—	28	16	no	—
122	C₂×C₄⋊C₄	—	2	2	no	—
123	C₂×C₄⋊C₄	—	7	4	no	—
124	C₂×C₄⋊C₄	—	14	16	no	—
125	C₂×C₄⋊C₄	—	14	2	no	—
126	C₂×C₄⋊C₄	C₂²×D₇	1	4	yes	—
127	C₂×C₄²	—	7	2	no	—
128	C₂×C₄²	—	7	2	no	—
129	C₂×C₄²	—	7	2	no	—
130	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
131	#107	—	2	8	no	—
132	#107	—	14	16	no	—
133	#107	—	14	8	no	—
134	C₄×C₈	—	7	16	no	—
135	C₄×C₈	—	7	4	no	—
136	C₄×C₈	C₂²⋊D₇	1	4	yes	—
137	C₂²×C₇	C₂×#127	1	1	yes	characteristic
138	C₂²×C₇	C₂×C₄⋊Q₈	1	2	yes	—
139	C₂²×C₇	C₂²×D₈	1	2	yes	—
140	C₂²×C₇	C₂²×Q₁₆	1	2	yes	—
141	C₄×C₇	—	8	4	no	—
142	C₄×C₇	#493	1	4	yes	—
143	C₄×C₇	#494	1	4	yes	—
144	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
145	C₄×C₇	C₂×C₄⋊Q₈	1	2	yes	—
146	Dic₇	—	8	4	no	—
147	Dic₇	#500	1	16	yes	complemented
148	C₂×Q₈	—	4	8	no	—
149	C₂×Q₈	—	4	4	no	—
150	C₂×Q₈	—	28	8	no	—
151	C₂×Q₈	—	28	4	no	—
152	C₂²×C₄	—	2	4	no	—
153	C₂²×C₄	—	7	4	no	—
154	C₂²×C₄	—	14	4	no	—
155	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
156	C₂²×C₄	D₄×D₇	1	1	yes	characteristic
157	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
158	C₂×C₈	—	7	4	no	—
159	C₂×C₈	—	7	8	no	—
160	C₂×C₈	Q₈×D₇	1	8	yes	—
161	C₂×C₈	C₂×C₂²⋊D₇	1	4	yes	—
162	C₄⋊C₄	—	2	4	no	—
163	C₄⋊C₄	—	2	4	no	—
164	C₄⋊C₄	—	4	4	no	—
165	C₄⋊C₄	—	14	4	no	—
166	C₄⋊C₄	—	14	4	no	—
167	C₄⋊C₄	—	28	32	no	—
168	C₄⋊C₄	—	28	4	no	—
169	C₄²	—	7	8	no	—
170	C₄²	—	7	8	no	—
171	C₄²	—	7	8	no	—
172	C₄²	D₄×D₇	1	2	yes	—
173	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
174	C₂×C₇	#2868	1	1	yes	characteristic
175	C₂×C₇	C₂×#493	1	1	yes	characteristic
176	C₂×C₇	C₂×#494	1	1	yes	characteristic
177	C₂×C₇	C₂×#500	1	4	yes	—
178	C₂³	C₂×D₄×D₇	1	1	yes	characteristic
179	Q₈	—	8	8	no	—
180	Q₈	—	56	8	no	—
181	C₂×C₄	—	4	8	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	7	8	no	—
184	C₂×C₄	—	7	8	no	—
185	C₂×C₄	—	7	8	no	—
186	C₂×C₄	—	28	8	no	—
187	C₂×C₄	—	28	4	no	—
188	C₂×C₄	D₇×D₈	1	4	yes	—
189	C₂×C₄	D₇×Q₁₆	1	4	yes	—
190	C₂×C₄	#6503	1	2	yes	—
191	C₂×C₄	#6503	1	2	yes	—
192	C₂×C₄	C₂×D₄×D₇	1	1	yes	characteristic
193	C₂×C₄	C₂×Q₈×D₇	1	4	yes	—
194	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
195	C₈	—	7	8	no	—
196	C₈	#6507	1	8	yes	complemented
197	C₇	#33312	1	1	yes	characteristic, complemented, Sylow
198	C₂²	D₇×C₄⋊Q₈	1	2	yes	—
199	C₂²	C₂×D₇×D₈	1	2	yes	—
200	C₂²	C₂×D₇×Q₁₆	1	2	yes	—
201	C₂²	C₂×#6503	1	1	yes	characteristic
202	C₄	—	7	16	no	—
203	C₄	—	8	4	no	—
204	C₄	—	56	4	no	—
205	C₄	#89923	1	4	yes	—
206	C₄	#89953	1	4	yes	—
207	C₄	D₇×C₄⋊Q₈	1	2	yes	—
208	C₄	C₂×#6507	1	2	yes	—
209	C₂	D₇×#500	1	4	yes	—
210	C₂	#11740698	1	1	yes	characteristic
211	C₂	C₂×#89923	1	1	yes	characteristic
212	C₂	C₂×#89953	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₇
commutator, cosocle	C₂²×C₄×C₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂×C₇×#500	C₂
socle, prime radical	C₂³×C₇	C₂²×D₄
hypercenter	C₂×#500	D₇
hypocenter	C₇	#33312
LCS	G ▷ C₂²×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
44. C₈×Dic₇	Q₈	semidirect
111. C₇×C₈	C₄⋊Q₈	semidirect
112. C₇⋊C₈	C₄⋊Q₈	semidirect
147. Dic₇	#500	semidirect
196. C₈	#6507	semidirect
197. C₇	#33312	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#500	2
#11740698	2
C₂×#89850	2
C₂×#89454	2
#11728360	2
C₄×C₈×Dic₇	2
#33312	7