Subgroups — #13272053

Order 1792.

470 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#89360	C₂	1	4	yes	maximal
3	#11727546	C₂	1	1	yes	maximal, characteristic
4	C₄×#6386	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇×Q₁₆	—	2	1	no	—
6	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊Q₈	—	2	1	no	—
8	C₇×#494	—	2	1	no	—
9	C₇×#494	—	2	1	no	—
10	C₇×#494	C₄	1	2	yes	—
11	C₇×#494	C₄	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	C₂×#6393	—	2	1	no	—
14	C₂×#6393	C₂²	1	2	yes	—
15	C₂×#6386	C₂²	1	2	yes	—
16	#89696	—	2	1	no	—
17	#89360	—	2	8	no	—
18	#89256	—	2	1	no	—
19	C₄×C₇⋊C₁₆	—	2	1	no	—
20	#13212	—	7	1	no	maximal, Sylow
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	2	4	no	—
25	C₂×C₇×Q₁₆	—	2	2	no	—
26	C₂×C₇×Q₁₆	—	2	4	no	—
27	C₂×C₇×Q₁₆	—	4	2	no	—
28	C₂×C₇×Q₁₆	—	4	2	no	—
29	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
32	C₇×C₄⋊Q₈	—	2	2	no	—
33	C₇×C₄⋊Q₈	—	2	2	no	—
34	C₇×C₄⋊Q₈	—	4	2	no	—
35	C₂×C₇×C₄⋊C₄	—	4	2	no	—
36	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
37	C₂×#1136	—	2	2	no	—
38	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄×C₇×C₈	D₄	1	1	yes	characteristic
41	C₄×C₇×C₈	D₄	1	1	yes	characteristic
42	#6393	—	2	4	no	—
43	#6393	—	4	2	no	—
44	#6386	—	2	2	no	—
45	#6386	D₄	1	4	yes	—
46	C₂×C₇⋊C₁₆	—	2	2	no	—
47	C₄⋊(C₇⋊C₈)	—	4	2	no	—
48	C₂×#494	D₇	1	1	yes	characteristic
49	C₂×#362	—	7	4	no	—
50	C₄×M₃₂	—	7	1	no	—
51	#1848	—	7	1	no	—
52	C₂×C₇×Q₈	—	2	4	no	—
53	C₂×C₇×Q₈	—	4	2	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂×C₇×Q₈	—	4	2	no	—
56	C₂×C₇×Q₈	—	4	2	no	—
57	C₂×C₇×Q₈	—	4	2	no	—
58	C₂²×C₄×C₇	—	4	2	no	—
59	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₇×Q₁₆	—	2	8	no	—
62	C₇×Q₁₆	—	4	4	no	—
63	C₇×Q₁₆	—	4	4	no	—
64	C₇×Q₁₆	—	4	4	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	D₈	1	2	yes	—
68	C₂×C₇×C₈	SD₁₆	1	2	yes	—
69	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₂×C₇×C₈	#41	1	2	yes	—
73	C₇×C₄⋊C₄	—	4	4	no	—
74	C₇×C₄⋊C₄	—	4	4	no	—
75	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₇	D₈	1	2	yes	—
78	#1136	—	4	2	no	—
79	#1136	—	4	2	no	—
80	C₂×C₇⋊C₈	—	4	2	no	—
81	C₇⋊C₁₆	—	2	4	no	—
82	C₂²×Q₁₆	—	2	1	no	—
83	C₂²×Q₁₆	C₂×D₇	1	2	yes	—
84	C₂×C₄⋊Q₈	—	2	1	no	—
85	C₂×M₃₂	—	7	2	no	—
86	#494	—	2	1	no	—
87	#494	—	2	1	no	—
88	#494	Dic₇	1	2	yes	—
89	#494	Dic₇	1	2	yes	—
90	C₂×#108	—	7	2	no	—
91	C₂×#108	—	14	1	no	—
92	#423	—	14	1	no	—
93	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
94	#362	—	14	8	no	—
95	#346	—	14	1	no	—
96	C₄×C₁₆	—	14	1	no	—
97	C₂³×C₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₇×Q₈	—	4	4	no	—
99	C₇×Q₈	—	4	4	no	—
100	C₇×Q₈	—	8	2	no	—
101	C₇×Q₈	—	8	2	no	—
102	C₂×C₄×C₇	—	2	4	no	—
103	C₂×C₄×C₇	—	4	4	no	—
104	C₂×C₄×C₇	—	8	2	no	—
105	C₂×C₄×C₇	—	8	2	no	—
106	C₂×C₄×C₇	#102	1	2	yes	—
107	C₂×C₄×C₇	#102	1	2	yes	—
108	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
109	C₂×C₄×C₇	#124	1	1	yes	characteristic
110	C₂×C₄×C₇	#127	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
114	C₇×C₈	#102	1	2	yes	—
115	C₇×C₈	#102	1	2	yes	—
116	C₇×C₈	C₄×D₄	1	2	yes	—
117	C₇×C₈	#121	1	2	yes	—
118	C₇⋊C₈	—	8	2	no	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂×Q₁₆	—	2	2	no	—
121	C₂×Q₁₆	—	2	2	no	—
122	C₂×Q₁₆	—	2	4	no	—
123	C₂×Q₁₆	—	2	4	no	—
124	C₂×Q₁₆	—	2	2	no	—
125	C₂×Q₁₆	—	4	2	no	—
126	C₂×Q₁₆	—	4	2	no	—
127	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
128	C₂×M₁₆	—	14	2	no	—
129	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	4	2	no	—
134	C₂×C₄⋊C₄	—	4	2	no	—
135	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
136	M₃₂	—	7	4	no	—
137	M₃₂	—	14	2	no	—
138	C₂×C₁₆	—	14	2	no	—
139	#108	—	14	4	no	—
140	#108	—	28	2	no	—
141	C₄⋊C₈	—	28	2	no	—
142	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
143	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
144	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
145	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
146	C₂²×C₇	—	2	2	no	—
147	C₂²×C₇	C₂×#102	1	1	yes	characteristic
148	C₂²×C₇	#486	1	1	yes	characteristic
149	C₂²×C₇	#492	1	1	yes	characteristic
150	C₄×C₇	—	2	2	no	—
151	C₄×C₇	—	8	4	no	—
152	C₄×C₇	#362	1	4	yes	complemented
153	C₄×C₇	#390	1	1	yes	characteristic
154	C₄×C₇	C₂×#102	1	1	yes	characteristic
155	C₄×C₇	#486	1	1	yes	characteristic
156	C₄×C₇	#493	1	1	yes	characteristic
157	C₂×Q₈	—	2	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	—	4	2	no	—
160	C₂×Q₈	—	4	2	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	#1142	1	2	yes	—
165	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
166	Q₁₆	—	2	8	no	—
167	Q₁₆	—	4	4	no	—
168	Q₁₆	—	4	4	no	—
169	Q₁₆	—	4	4	no	—
170	M₁₆	—	28	2	no	—
171	M₁₆	—	28	2	no	—
172	C₂×C₈	—	28	2	no	—
173	C₂×C₈	#1141	1	2	yes	—
174	C₂×C₈	#1142	1	2	yes	—
175	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
176	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
177	C₂×C₈	D₄×D₇	1	2	yes	—
178	C₂×C₈	#1159	1	2	yes	—
179	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
180	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
181	C₄⋊C₄	—	4	4	no	—
182	C₄⋊C₄	—	4	4	no	—
183	C₄²	#1141	1	2	yes	—
184	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
185	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
186	C₁₆	—	14	4	no	—
187	C₂×C₇	—	2	1	no	—
188	C₂×C₇	—	2	1	no	—
189	C₂×C₇	#1957	1	1	yes	characteristic
190	C₂×C₇	C₂×#362	1	2	yes	—
191	C₂³	C₂×#1142	1	1	yes	characteristic
192	Q₈	—	4	4	no	—
193	Q₈	—	4	4	no	—
194	Q₈	—	8	2	no	—
195	Q₈	—	8	2	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	4	4	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	2	no	—
200	C₂×C₄	#6406	1	2	yes	—
201	C₂×C₄	#6406	1	2	yes	—
202	C₂×C₄	C₂×#1141	1	1	yes	characteristic
203	C₂×C₄	C₂×#1141	1	1	yes	characteristic
204	C₂×C₄	C₂×#1142	1	1	yes	characteristic
205	C₂×C₄	#6499	1	1	yes	characteristic
206	C₂×C₄	#6503	1	1	yes	characteristic
207	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
208	C₈	—	56	2	no	—
209	C₈	#6406	1	2	yes	—
210	C₈	#6406	1	2	yes	—
211	C₈	D₄×Dic₇	1	2	yes	—
212	C₈	#6501	1	2	yes	—
213	C₇	#13212	1	1	yes	characteristic, complemented, Sylow
214	C₂²	—	2	2	no	—
215	C₂²	#89842	1	1	yes	characteristic
216	C₂²	#89848	1	1	yes	characteristic
217	C₂²	C₂×#6406	1	1	yes	characteristic
218	C₄	—	2	2	no	—
219	C₄	—	8	4	no	—
220	C₄	#89360	1	4	yes	complemented
221	C₄	#89842	1	1	yes	characteristic
222	C₄	#89845	1	1	yes	characteristic
223	C₄	C₂×#6406	1	1	yes	characteristic
224	C₄	#89993	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	1	no	—
227	C₂	#11729518	1	1	yes	characteristic
228	C₂	C₂×#89360	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#494	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	C₂×SD₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#494	D₇
hypocenter	C₇	#13212
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
152. C₄×C₇	#362	semidirect
213. C₇	#13212	semidirect
220. C₄	#89360	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#494	2
C₂×#89360	2
#11727546	2
C₄×#6386	2
#13212	7