Subgroups — #13268658

Order 1792.

586 conjugacy classes of subgroups.

The subgroup lattice has 234 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	#11725470	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	2	no	—
8	C₇×#405	—	2	2	no	—
9	C₂×C₇×#97	C₂²	1	2	yes	—
10	C₂×C₄×C₇×C₈	C₄	1	2	yes	—
11	C₂×#6386	—	2	4	no	—
12	C₂×#6386	—	2	2	no	—
13	C₂×#6386	C₄	1	8	yes	complemented
14	C₂×#6386	C₂²	1	4	yes	—
15	C₂²×C₇⋊C₁₆	—	2	2	no	—
16	#12551	—	7	1	no	maximal, Sylow
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	C₈	1	4	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
22	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₇×C₈	D₄	1	2	yes	—
25	C₂²×C₇×C₈	D₄	1	2	yes	—
26	C₂²×C₇×C₈	D₄	1	2	yes	—
27	C₂²×C₇×C₈	Q₈	1	2	yes	—
28	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
29	C₇×#117	—	2	2	no	—
30	C₇×#117	—	2	2	no	—
31	C₂×C₇×C₄⋊C₄	C₈	1	2	yes	—
32	C₂×C₇×C₂²⋊C₄	C₈	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₇×#97	—	2	4	no	—
35	C₄×C₇×C₈	—	2	2	no	—
36	C₄×C₇×C₈	—	2	2	no	—
37	#6386	—	2	8	no	—
38	#6386	—	2	4	no	—
39	#6386	C₂×C₄	1	8	yes	—
40	C₂×C₇⋊C₁₆	—	2	8	no	—
41	C₂×C₇⋊C₁₆	—	2	4	no	—
42	C₂²×M₃₂	—	7	2	no	—
43	C₂×#405	D₇	1	1	yes	characteristic
44	#1377	—	7	4	no	—
45	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₄²	1	2	yes	—
47	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
48	C₂²×C₄×C₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
50	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
52	C₂²×C₄×C₇	M₁₆	1	2	yes	—
53	C₂²×C₄×C₇	M₁₆	1	2	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₇×M₁₆	—	2	8	no	—
56	C₂×C₇×C₈	—	2	2	no	—
57	C₂×C₇×C₈	—	2	2	no	—
58	C₂×C₇×C₈	C₄²	1	2	yes	—
59	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
62	C₂×C₇×C₈	C₂²⋊C₄	1	4	yes	—
63	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
64	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
65	C₂×C₇×C₈	C₄⋊C₄	1	4	yes	—
66	C₂×C₇×C₈	C₂×C₈	1	4	yes	—
67	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
68	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
69	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
70	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₂²⋊C₄	—	2	4	no	—
73	C₄²×C₇	—	2	2	no	—
74	C₄²×C₇	—	2	2	no	—
75	C₇⋊C₁₆	—	2	8	no	—
76	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
77	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
78	C₂×#117	Dic₇	1	1	yes	characteristic
79	C₂×M₃₂	—	7	8	no	—
80	C₂×M₃₂	—	7	4	no	—
81	C₂×M₃₂	—	14	4	no	—
82	C₂×M₃₂	—	14	2	no	—
83	C₂²×C₁₆	—	14	2	no	—
84	#405	—	2	2	no	—
85	#405	—	2	2	no	—
86	C₂×#97	C₂×D₇	1	2	yes	—
87	C₂×C₄×C₈	Dic₇	1	2	yes	—
88	C₂³×C₇	—	4	2	no	—
89	C₂³×C₇	C₄×C₈	1	2	yes	—
90	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
91	C₂³×C₇	C₄⋊C₈	1	2	yes	—
92	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₇	—	2	4	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	#95	1	4	yes	—
96	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₇	#97	1	4	yes	—
98	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
99	C₂×C₄×C₇	C₂²⋊C₈	1	4	yes	—
100	C₂×C₄×C₇	C₄⋊C₈	1	2	yes	—
101	C₂×C₄×C₇	C₄⋊C₈	1	4	yes	—
102	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
108	C₇×C₈	—	2	4	no	—
109	C₇×C₈	#95	1	2	yes	—
110	C₇×C₈	#95	1	2	yes	—
111	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
112	C₇×C₈	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
114	C₂×M₁₆	—	2	2	no	—
115	C₂×M₁₆	—	2	2	no	—
116	C₂×M₁₆	C₇⋊C₈	1	4	yes	—
117	C₂²×C₈	Dic₁₄	1	2	yes	—
118	C₂²×C₈	C₄×D₇	1	4	yes	—
119	C₂²×C₈	D₂₈	1	2	yes	—
120	C₂²×C₈	C₂×Dic₇	1	2	yes	—
121	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
123	C₂²×C₈	C₂²⋊D₇	1	2	yes	—
124	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
125	#117	—	2	2	no	—
126	#117	—	2	2	no	—
127	C₂×C₄⋊C₄	C₇⋊C₈	1	2	yes	—
128	C₂×C₂²⋊C₄	C₇⋊C₈	1	2	yes	—
129	C₂×C₄²	C₂×Dic₇	1	2	yes	—
130	M₃₂	—	7	8	no	—
131	M₃₂	—	14	8	no	—
132	M₃₂	—	14	4	no	—
133	C₂×C₁₆	—	14	8	no	—
134	C₂×C₁₆	—	14	4	no	—
135	#97	—	2	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	4	no	—
140	C₂²×C₇	—	4	2	no	—
141	C₂²×C₇	—	4	2	no	—
142	C₂²×C₇	#336	1	4	yes	—
143	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
146	C₄×C₇	—	4	4	no	—
147	C₄×C₇	#336	1	4	yes	—
148	C₄×C₇	C₂×#95	1	1	yes	characteristic
149	C₄×C₇	C₂×#97	1	1	yes	characteristic
150	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
152	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
153	C₂²×C₄	C₂×C₇⋊C₈	1	4	yes	—
154	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
155	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
156	C₂²×C₄	#1136	1	2	yes	—
157	C₂²×C₄	#1136	1	2	yes	—
158	C₂²×C₄	C₄×Dic₇	1	2	yes	—
159	C₂²×C₄	C₄⋊Dic₇	1	2	yes	—
160	C₂²×C₄	C₂²⋊Dic₇	1	2	yes	—
161	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
162	M₁₆	—	2	8	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	C₂×C₇⋊C₈	1	4	yes	—
166	C₂×C₈	C₄×Dic₇	1	2	yes	—
167	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
168	C₂×C₈	Dic₇⋊C₄	1	4	yes	—
169	C₂×C₈	C₄⋊Dic₇	1	2	yes	—
170	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
171	C₂×C₈	(C₂×D₇)⋊C₄	1	4	yes	—
172	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
174	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
176	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
178	C₄⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	4	no	—
180	C₄²	—	2	2	no	—
181	C₄²	—	2	2	no	—
182	C₁₆	—	14	8	no	—
183	C₂×C₇	—	2	4	no	—
184	C₂×C₇	—	2	1	no	—
185	C₂×C₇	—	2	1	no	—
186	C₂×C₇	C₂×#336	1	1	yes	characteristic
187	C₂×C₇	#2115	1	2	yes	—
188	C₂³	—	4	2	no	—
189	C₂³	C₄×C₇⋊C₈	1	2	yes	—
190	C₂³	C₄⋊(C₇⋊C₈)	1	2	yes	—
191	C₂³	C₂²⋊(C₇⋊C₈)	1	2	yes	—
192	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	4	4	no	—
195	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
196	C₂×C₄	#6377	1	4	yes	—
197	C₂×C₄	C₄⋊(C₇⋊C₈)	1	4	yes	—
198	C₂×C₄	C₄⋊(C₇⋊C₈)	1	2	yes	—
199	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	4	yes	—
200	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
201	C₂×C₄	C₇⋊#95	1	4	yes	—
202	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
203	C₂×C₄	C₂×#1136	1	1	yes	characteristic
204	C₂×C₄	C₂×#1136	1	1	yes	characteristic
205	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
206	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
208	C₈	—	2	4	no	—
209	C₈	C₇⋊#95	1	2	yes	—
210	C₈	C₇⋊#95	1	2	yes	—
211	C₈	C₂×Dic₇⋊C₄	1	2	yes	—
212	C₈	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
213	C₇	#12551	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	—	2	4	no	—
216	C₂²	—	4	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	#89319	1	4	yes	—
219	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
220	C₂²	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
221	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
222	C₄	—	4	4	no	—
223	C₄	#89319	1	4	yes	—
224	C₄	C₂×#6377	1	1	yes	characteristic
225	C₄	C₂×C₄⋊(C₇⋊C₈)	1	1	yes	characteristic
226	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
227	C₄	C₂×C₇⋊#95	1	1	yes	characteristic
228	C₂	—	2	1	no	—
229	C₂	—	2	1	no	—
230	C₂	—	2	4	no	—
231	C₂	C₂×#89319	1	1	yes	characteristic
232	C₂	#11729761	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#405	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#405	D₇
hypocenter	C₇	#12551
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
3. #11725470	C₂	semidirect
13. C₂×#6386	C₄	semidirect
213. C₇	#12551	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
#11725470	2
#12551	7