Subgroups — #13268646

Order 1792.

586 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×#405	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6386	C₂	1	2	yes	maximal
3	#11725469	C₂	1	4	yes	maximal, complemented
4	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₇×#117	C₄	1	1	yes	characteristic
7	C₇×#405	—	2	4	no	—
8	C₂×C₇×#97	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
10	C₂×#6386	—	2	4	no	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	C₄	1	4	yes	complemented
13	C₂×#6386	C₂²	1	4	yes	—
14	C₂²×C₇⋊C₁₆	C₄	1	4	yes	complemented
15	#12547	—	7	1	no	maximal, Sylow
16	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₇×M₁₆	—	2	2	no	—
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂×C₇×M₁₆	C₈	1	4	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
24	C₂²×C₇×C₈	D₄	1	2	yes	—
25	C₂²×C₇×C₈	D₄	1	4	yes	—
26	C₂²×C₇×C₈	Q₈	1	2	yes	—
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₇×#117	—	2	2	no	—
29	C₇×#117	C₈	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	—	2	1	no	—
31	C₂×C₇×C₂²⋊C₄	—	2	1	no	—
32	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
33	C₇×#97	—	2	4	no	—
34	C₄×C₇×C₈	—	2	4	no	—
35	#6386	—	2	8	no	—
36	#6386	—	2	4	no	—
37	#6386	—	2	4	no	—
38	C₂×C₇⋊C₁₆	—	2	8	no	—
39	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	—
40	C₂²×M₃₂	—	7	2	no	—
41	C₂×#405	D₇	1	1	yes	characteristic
42	#1375	—	7	4	no	—
43	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇	—	2	2	no	—
45	C₂²×C₄×C₇	C₄²	1	2	yes	—
46	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
47	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
50	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
51	C₂²×C₄×C₇	M₁₆	1	2	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
53	C₇×M₁₆	—	2	8	no	—
54	C₂×C₇×C₈	—	2	2	no	—
55	C₂×C₇×C₈	—	2	2	no	—
56	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
57	C₂×C₇×C₈	C₄²	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
59	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
60	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
61	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
62	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
63	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
64	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
65	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
66	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
68	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
69	C₇×C₄⋊C₄	—	2	4	no	—
70	C₇×C₂²⋊C₄	—	2	4	no	—
71	C₄²×C₇	—	2	2	no	—
72	C₄²×C₇	C₂×C₈	1	4	yes	—
73	C₇⋊C₁₆	—	2	8	no	—
74	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
75	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
76	C₂×#117	Dic₇	1	1	yes	characteristic
77	C₂×M₃₂	—	7	4	no	—
78	C₂×M₃₂	—	7	4	no	—
79	C₂×M₃₂	—	14	4	no	—
80	C₂×M₃₂	—	14	4	no	—
81	C₂²×C₁₆	—	7	4	no	—
82	#405	—	2	4	no	—
83	C₂×#97	C₂×D₇	1	2	yes	—
84	C₂×C₄×C₈	Dic₇	1	2	yes	—
85	C₂³×C₇	—	4	2	no	—
86	C₂³×C₇	#97	1	2	yes	—
87	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
88	C₂³×C₇	C₄⋊C₈	1	2	yes	—
89	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	#95	1	4	yes	—
93	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
94	C₂×C₄×C₇	#97	1	2	yes	—
95	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
97	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
98	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
99	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
105	C₇×C₈	—	2	4	no	—
106	C₇×C₈	#95	1	2	yes	—
107	C₇×C₈	#95	1	2	yes	—
108	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
109	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
110	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
111	C₂×M₁₆	—	2	2	no	—
112	C₂×M₁₆	—	2	2	no	—
113	C₂×M₁₆	C₇⋊C₈	1	4	yes	—
114	C₂²×C₈	Dic₁₄	1	2	yes	—
115	C₂²×C₈	C₄×D₇	1	2	yes	—
116	C₂²×C₈	C₄×D₇	1	2	yes	—
117	C₂²×C₈	D₂₈	1	2	yes	—
118	C₂²×C₈	C₂×Dic₇	1	2	yes	—
119	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
120	C₂²×C₈	C₂²⋊D₇	1	4	yes	—
121	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
122	#117	—	2	2	no	—
123	#117	C₇⋊C₈	1	4	yes	—
124	C₂×C₄⋊C₄	—	2	1	no	—
125	C₂×C₂²⋊C₄	—	2	1	no	—
126	C₂×C₄²	C₂×Dic₇	1	2	yes	—
127	M₃₂	—	14	8	no	—
128	M₃₂	—	14	4	no	—
129	M₃₂	—	14	4	no	—
130	C₂×C₁₆	—	7	8	no	—
131	C₂×C₁₆	—	14	8	no	—
132	#97	—	2	4	no	—
133	C₄×C₈	—	2	4	no	—
134	C₂²×C₇	—	2	2	no	—
135	C₂²×C₇	—	2	4	no	—
136	C₂²×C₇	—	4	2	no	—
137	C₂²×C₇	—	4	2	no	—
138	C₂²×C₇	#336	1	4	yes	—
139	C₂²×C₇	C₂×#97	1	1	yes	characteristic
140	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
141	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
142	C₄×C₇	—	4	4	no	—
143	C₄×C₇	#336	1	4	yes	—
144	C₄×C₇	C₂×#95	1	1	yes	characteristic
145	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
146	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₂⁴	#1136	1	1	yes	characteristic
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
151	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
152	C₂²×C₄	#1136	1	2	yes	—
153	C₂²×C₄	#1136	1	1	yes	characteristic
154	C₂²×C₄	C₄×Dic₇	1	2	yes	—
155	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
156	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
157	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
158	M₁₆	—	2	8	no	—
159	C₂×C₈	—	2	2	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
162	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
163	C₂×C₈	C₄×Dic₇	1	1	yes	characteristic
164	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
165	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
166	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
167	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
168	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
169	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
170	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
171	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
172	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
173	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
174	C₄⋊C₄	—	2	4	no	—
175	C₂²⋊C₄	—	2	4	no	—
176	C₄²	—	2	2	no	—
177	C₄²	C₂×C₇⋊C₈	1	4	yes	—
178	C₁₆	—	14	8	no	—
179	C₂×C₇	—	2	1	no	—
180	C₂×C₇	—	2	4	no	—
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	C₂×#336	1	1	yes	characteristic
183	C₂×C₇	#2110	1	2	yes	—
184	C₂³	—	4	2	no	—
185	C₂³	#6377	1	2	yes	—
186	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
187	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
188	C₂³	C₂×#1136	1	1	yes	characteristic
189	C₂×C₄	—	2	4	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	C₄×C₇⋊C₈	1	4	yes	—
192	C₂×C₄	#6377	1	2	yes	—
193	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
194	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
195	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
196	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
197	C₂×C₄	C₇⋊#95	1	4	yes	—
198	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
199	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₂×#1136	1	1	yes	characteristic
201	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
202	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
204	C₈	—	2	4	no	—
205	C₈	C₇⋊#95	1	2	yes	—
206	C₈	C₇⋊#95	1	2	yes	—
207	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
208	C₈	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
209	C₇	#12547	1	1	yes	characteristic, complemented, Sylow
210	C₂²	—	2	2	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	4	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	#89319	1	4	yes	—
215	C₂²	C₂×#6377	1	1	yes	characteristic
216	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
217	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
218	C₄	—	4	4	no	—
219	C₄	#89319	1	4	yes	—
220	C₄	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
221	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
222	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
223	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
224	C₂	—	2	1	no	—
225	C₂	—	2	1	no	—
226	C₂	—	2	4	no	—
227	C₂	#11727638	1	2	yes	—
228	C₂	C₂×#89319	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂×#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#12547
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #11725469	C₂	semidirect
12. C₂×#6386	C₄	semidirect
14. C₂²×C₇⋊C₁₆	C₄	semidirect
209. C₇	#12547	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#405	2
C₂²×#6386	2
#11725469	2
#12547	7