Subgroups — #13269106

Order 1792.

414 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₇⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#89256	C₂	1	1	yes	maximal, characteristic
5	#11725610	C₂	1	2	yes	maximal
6	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
13	C₄²×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₇×#346	—	2	2	no	—
16	Dic₇⋊C₁₆	C₄	1	8	yes	complemented
17	#89256	—	2	2	no	—
18	#12653	—	7	1	no	maximal, Sylow
19	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₇	C₈	1	2	yes	—
23	C₂×C₄×Dic₇	C₈	1	2	yes	—
24	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂×C₇×C₁₆	—	2	2	no	—
29	C₂×C₇×C₁₆	—	2	2	no	—
30	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	—
31	C₄×C₇×C₈	D₄	1	2	yes	—
32	C₄×C₇×C₈	Q₈	1	2	yes	—
33	C₈×Dic₇	—	2	2	no	—
34	C₈×Dic₇	—	2	2	no	—
35	C₈×Dic₇	C₈	1	4	yes	—
36	C₈×Dic₇	C₂×C₄	1	4	yes	—
37	C₂×C₇⋊C₁₆	—	2	2	no	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×C₇⋊C₁₆	C₂×C₄	1	4	yes	—
40	C₄×C₇⋊C₈	—	2	2	no	—
41	C₂×C₄⋊C₁₆	—	7	2	no	—
42	C₂×#346	—	7	1	no	—
43	C₂×#346	D₇	1	1	yes	characteristic
44	C₄²×C₈	—	7	1	no	—
45	#1374	—	7	2	no	—
46	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
47	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
49	C₂²×Dic₇	C₂×C₈	1	2	yes	—
50	C₂²×Dic₇	C₂×C₈	1	1	yes	characteristic
51	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
52	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
53	C₂×C₇×C₈	M₁₆	1	2	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
55	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇×C₈	#41	1	2	yes	—
58	C₄²×C₇	C₄⋊C₄	1	2	yes	—
59	C₄²×C₇	M₁₆	1	2	yes	—
60	C₄×Dic₇	—	2	1	no	—
61	C₄×Dic₇	—	2	1	no	—
62	C₄×Dic₇	—	2	4	no	—
63	C₄×Dic₇	—	2	4	no	—
64	C₄×Dic₇	C₄²	1	2	yes	—
65	C₄×Dic₇	C₂×C₈	1	2	yes	—
66	C₂×C₇⋊C₈	—	2	1	no	—
67	C₂×C₇⋊C₈	—	2	1	no	—
68	C₂×C₇⋊C₈	—	2	1	no	—
69	C₂×C₇⋊C₈	—	2	1	no	—
70	C₂×C₇⋊C₈	C₄²	1	2	yes	—
71	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
72	C₇×C₁₆	—	2	4	no	—
73	C₇⋊C₁₆	—	2	4	no	—
74	C₂²×C₁₆	—	7	2	no	—
75	C₂²×C₁₆	C₂×D₇	1	2	yes	—
76	C₂×C₄×C₈	—	7	1	no	—
77	C₂×C₄×C₈	—	7	1	no	—
78	C₂×C₄×C₈	—	7	1	no	—
79	C₂×C₄×C₈	—	7	1	no	—
80	C₂×C₄×C₈	—	7	1	no	—
81	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
82	C₄³	—	7	1	no	—
83	C₄⋊C₁₆	—	7	8	no	—
84	#346	—	2	2	no	—
85	#346	—	14	2	no	—
86	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
90	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
91	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₇	#117	1	1	yes	characteristic
93	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
94	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
96	C₂×Dic₇	—	2	2	no	—
97	C₂×Dic₇	—	2	2	no	—
98	C₂×Dic₇	—	2	2	no	—
99	C₂×Dic₇	C₄×C₈	1	2	yes	—
100	C₂×Dic₇	C₄×C₈	1	2	yes	—
101	C₂×Dic₇	M₃₂	1	4	yes	—
102	C₂×Dic₇	M₃₂	1	4	yes	—
103	C₇×C₈	C₄⋊C₈	1	4	yes	—
104	C₇×C₈	C₄×D₄	1	2	yes	—
105	C₇×C₈	C₄×Q₈	1	2	yes	—
106	C₇⋊C₈	—	2	2	no	—
107	C₇⋊C₈	—	2	2	no	—
108	C₂²×C₈	—	7	1	no	—
109	C₂²×C₈	—	7	1	no	—
110	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
111	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
112	C₂×C₄²	—	7	1	no	—
113	C₂×C₄²	—	7	2	no	—
114	C₂×C₄²	—	7	2	no	—
115	C₂×C₄²	—	7	1	no	—
116	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
117	C₂×C₁₆	—	2	2	no	—
118	C₂×C₁₆	—	2	2	no	—
119	C₂×C₁₆	—	7	4	no	—
120	C₂×C₁₆	—	14	2	no	—
121	C₂×C₁₆	—	14	2	no	—
122	C₂×C₁₆	C₄×D₇	1	4	yes	—
123	C₄×C₈	—	7	4	no	—
124	C₄×C₈	—	7	4	no	—
125	C₄×C₈	—	14	2	no	—
126	C₄×C₈	—	14	2	no	—
127	C₄×C₈	—	14	2	no	—
128	C₄×C₈	Dic₁₄	1	2	yes	—
129	C₄×C₈	C₂²⋊D₇	1	2	yes	—
130	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
131	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
132	C₂²×C₇	#431	1	1	yes	characteristic
133	C₂²×C₇	C₂×M₃₂	1	2	yes	—
134	C₂²×C₇	C₂×M₃₂	1	2	yes	—
135	C₄×C₇	—	2	4	no	—
136	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
137	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
138	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
139	C₄×C₇	#431	1	1	yes	characteristic
140	Dic₇	—	2	2	no	—
141	Dic₇	—	2	4	no	—
142	Dic₇	#346	1	4	yes	complemented
143	C₂²×C₄	—	7	1	no	—
144	C₂²×C₄	—	7	2	no	—
145	C₂²×C₄	—	7	1	no	—
146	C₂²×C₄	C₈×D₇	1	2	yes	—
147	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
148	C₂×C₈	—	7	2	no	—
149	C₂×C₈	—	7	2	no	—
150	C₂×C₈	—	14	1	no	—
151	C₂×C₈	—	14	1	no	—
152	C₂×C₈	—	14	1	no	—
153	C₂×C₈	—	14	1	no	—
154	C₂×C₈	C₈×D₇	1	2	yes	—
155	C₂×C₈	#1131	1	2	yes	—
156	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
157	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
158	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
159	C₂×C₈	#1157	1	2	yes	—
160	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
161	C₄²	—	7	2	no	—
162	C₄²	—	7	2	no	—
163	C₄²	—	14	4	no	—
164	C₄²	—	14	4	no	—
165	C₄²	—	14	1	no	—
166	C₄²	—	14	1	no	—
167	C₄²	#1131	1	2	yes	—
168	C₄²	Dic₇⋊C₄	1	2	yes	—
169	C₁₆	—	2	4	no	—
170	C₁₆	—	14	4	no	—
171	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
172	C₂×C₇	C₂×#346	1	1	yes	characteristic
173	C₂×C₇	C₄×M₃₂	1	1	yes	characteristic
174	C₂×C₇	#2150	1	2	yes	—
175	C₂×C₇	#2163	1	2	yes	—
176	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	7	4	no	—
180	C₂×C₄	—	7	2	no	—
181	C₂×C₄	—	7	4	no	—
182	C₂×C₄	—	7	2	no	—
183	C₂×C₄	—	14	2	no	—
184	C₂×C₄	—	14	2	no	—
185	C₂×C₄	—	14	2	no	—
186	C₂×C₄	Dic₇⋊C₈	1	4	yes	—
187	C₂×C₄	C₄²×D₇	1	1	yes	characteristic
188	C₂×C₄	#6435	1	1	yes	characteristic
189	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
190	C₂×C₄	C₂×#1131	1	1	yes	characteristic
191	C₂×C₄	C₂×#1131	1	1	yes	characteristic
192	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
193	C₈	—	14	2	no	—
194	C₈	—	14	2	no	—
195	C₈	Dic₇⋊C₈	1	4	yes	—
196	C₈	C₄×Dic₁₄	1	2	yes	—
197	C₈	C₄×C₂²⋊D₇	1	2	yes	—
198	C₇	#12653	1	1	yes	characteristic, complemented, Sylow
199	C₂²	C₄×C₈×D₇	1	1	yes	characteristic
200	C₂²	#89457	1	1	yes	characteristic
201	C₂²	D₇×M₃₂	1	2	yes	—
202	C₂²	D₇×M₃₂	1	2	yes	—
203	C₂²	C₂×Dic₇⋊C₈	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	7	4	no	—
206	C₄	—	14	2	no	—
207	C₄	—	14	4	no	—
208	C₄	#89457	1	1	yes	characteristic
209	C₄	C₄×#1131	1	1	yes	characteristic
210	C₄	C₄×Dic₇⋊C₄	1	1	yes	characteristic
211	C₄	C₂×Dic₇⋊C₈	1	1	yes	characteristic
212	C₂	D₇×#346	1	1	yes	characteristic
213	C₂	#11726825	1	1	yes	characteristic
214	C₂	#11726827	1	2	yes	—
215	C₂	C₄×Dic₇⋊C₈	1	1	yes	characteristic
216	C₂	#11729174	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇
commutator	C₂²×C₇	C₂×C₄×C₈
Fitting	C₂×C₇×#346	C₂
socle, prime radical	C₂³×C₇	C₂²×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#346	D₇
hypocenter	C₇	#12653
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
16. Dic₇⋊C₁₆	C₄	semidirect
142. Dic₇	#346	semidirect
198. C₇	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#346	2
C₂×Dic₇⋊C₁₆	2
C₄×C₈×Dic₇	2
C₂×#89256	2
#11725610	2
#12653	7