Subgroups — #13265188

Order 1792.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 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	#11727498	C₂	1	1	yes	maximal, characteristic
3	#11727485	C₂	1	1	yes	maximal, characteristic
4	#11725599	C₂	1	4	yes	maximal, complemented
5	C₂²×C₇×Q₁₆	—	2	1	no	—
6	C₂²×C₇×Q₁₆	C₄	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	C₂²	1	2	yes	—
12	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
13	#89696	—	2	1	no	—
14	C₂²⋊(C₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₇⋊C₁₆)	C₂²	1	2	yes	—
16	C₄×C₇⋊C₁₆	—	2	1	no	—
17	#89253	—	2	1	no	—
18	#89253	C₂²	1	2	yes	—
19	#11962	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	2	2	no	—
21	C₂×C₇×Q₁₆	—	2	2	no	—
22	C₂×C₇×Q₁₆	—	2	2	no	—
23	C₂×C₇×Q₁₆	—	2	2	no	—
24	C₂×C₇×Q₁₆	—	4	4	no	—
25	C₂×C₇×Q₁₆	—	4	4	no	—
26	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
29	C₇×C₄⋊Q₈	—	2	2	no	—
30	C₇×C₄⋊Q₈	—	2	2	no	—
31	C₇×C₄⋊Q₈	—	4	1	no	—
32	C₇×C₄⋊Q₈	—	4	1	no	—
33	C₂×C₇×C₄⋊C₄	—	4	1	no	—
34	C₂×C₇×C₄⋊C₄	—	4	1	no	—
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×#1136	—	4	1	no	—
37	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
38	C₄×C₇×C₈	D₄	1	1	yes	characteristic
39	C₄×C₇×C₈	D₄	1	1	yes	characteristic
40	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
41	C₂×C₇⋊C₁₆	—	2	2	no	—
42	C₄⋊(C₇⋊C₈)	—	2	2	no	—
43	C₄⋊(C₇⋊C₈)	—	4	1	no	—
44	C₂×#494	D₇	1	1	yes	characteristic
45	#2162	—	7	1	no	—
46	#1567	—	7	1	no	—
47	#1362	—	7	4	no	—
48	C₂×C₇×Q₈	—	2	4	no	—
49	C₂×C₇×Q₈	—	4	4	no	—
50	C₂×C₇×Q₈	—	4	4	no	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂²×C₄×C₇	—	4	2	no	—
53	C₂²×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₇	SD₁₆	1	1	yes	characteristic
56	C₇×Q₁₆	—	4	4	no	—
57	C₇×Q₁₆	—	4	4	no	—
58	C₇×Q₁₆	—	4	4	no	—
59	C₇×Q₁₆	—	4	4	no	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
66	C₇×C₄⋊C₄	—	4	2	no	—
67	C₇×C₄⋊C₄	—	4	2	no	—
68	C₇×C₄⋊C₄	—	4	2	no	—
69	C₇×C₄⋊C₄	—	4	2	no	—
70	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₇	D₈	1	1	yes	characteristic
72	C₄²×C₇	SD₁₆	1	1	yes	characteristic
73	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
74	#1136	—	8	2	no	—
75	C₂×C₇⋊C₈	—	4	2	no	—
76	C₇⋊C₁₆	—	4	2	no	—
77	C₂²×Q₁₆	—	2	1	no	—
78	C₂²×Q₁₆	Dic₇	1	2	yes	—
79	C₂×C₄⋊Q₈	—	2	1	no	—
80	#494	—	2	1	no	—
81	#494	—	2	1	no	—
82	#494	Dic₇	1	2	yes	—
83	#494	C₂×D₇	1	2	yes	—
84	#423	—	14	1	no	—
85	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
86	C₄⋊C₁₆	—	7	2	no	—
87	C₂²⋊C₁₆	—	14	1	no	—
88	C₄×C₁₆	—	14	1	no	—
89	#335	—	7	2	no	—
90	#335	—	14	1	no	—
91	C₂³×C₇	#102	1	1	yes	characteristic
92	C₇×Q₈	—	4	4	no	—
93	C₇×Q₈	—	4	4	no	—
94	C₇×Q₈	—	8	4	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	—	2	1	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	—	8	4	no	—
100	C₂×C₄×C₇	#102	1	1	yes	characteristic
101	C₂×C₄×C₇	#102	1	1	yes	characteristic
102	C₂×C₄×C₇	#102	1	1	yes	characteristic
103	C₂×C₄×C₇	SD₃₂	1	2	yes	—
104	C₂×C₄×C₇	Q₃₂	1	2	yes	—
105	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
108	C₇×C₈	—	2	1	no	—
109	C₇×C₈	—	2	1	no	—
110	C₇×C₈	#100	1	2	yes	—
111	C₇×C₈	AGL(1,ℤ/8)	1	2	yes	—
112	C₇⋊C₈	—	8	2	no	—
113	C₂²×Q₈	—	2	2	no	—
114	C₂×Q₁₆	—	2	2	no	—
115	C₂×Q₁₆	—	2	2	no	—
116	C₂×Q₁₆	—	2	2	no	—
117	C₂×Q₁₆	—	4	4	no	—
118	C₂×Q₁₆	—	4	4	no	—
119	C₂×Q₁₆	C₂×Dic₇	1	4	yes	—
120	C₂×M₁₆	—	28	1	no	—
121	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
122	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
123	C₄⋊Q₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	2	no	—
125	C₄⋊Q₈	—	4	1	no	—
126	C₄⋊Q₈	—	4	1	no	—
127	C₂×C₄⋊C₄	—	4	1	no	—
128	C₂×C₄⋊C₄	—	4	1	no	—
129	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
130	C₂×C₁₆	—	14	2	no	—
131	C₄⋊C₈	—	14	2	no	—
132	C₄⋊C₈	—	28	1	no	—
133	C₄×C₈	C₂×Dic₇	1	1	yes	characteristic
134	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
135	C₄×C₈	C₂²⋊D₇	1	1	yes	characteristic
136	C₄×C₈	C₂²×D₇	1	1	yes	characteristic
137	C₂²×C₇	—	2	1	no	—
138	C₂²×C₇	—	2	1	no	—
139	C₂²×C₇	#358	1	2	yes	—
140	C₂²×C₇	C₂×#102	1	1	yes	characteristic
141	C₄×C₇	—	2	1	no	—
142	C₄×C₇	—	2	1	no	—
143	C₄×C₇	—	8	4	no	—
144	C₄×C₇	#358	1	2	yes	—
145	C₄×C₇	#362	1	2	yes	—
146	C₄×C₇	C₂×#100	1	1	yes	characteristic
147	C₄×C₇	#418	1	1	yes	characteristic
148	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
149	C₄×C₇	C₂×Q₃₂	1	1	yes	characteristic
150	C₂×Q₈	—	2	4	no	—
151	C₂×Q₈	—	4	4	no	—
152	C₂×Q₈	—	4	4	no	—
153	C₂×Q₈	—	4	2	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	#1141	1	1	yes	characteristic
156	C₂²×C₄	#1142	1	1	yes	characteristic
157	C₂²×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
158	Q₁₆	—	4	4	no	—
159	Q₁₆	—	4	4	no	—
160	Q₁₆	—	4	4	no	—
161	Q₁₆	—	4	4	no	—
162	M₁₆	—	56	2	no	—
163	C₂×C₈	—	2	2	no	—
164	C₂×C₈	—	2	2	no	—
165	C₂×C₈	—	28	2	no	—
166	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
167	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
168	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
169	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
170	C₄⋊C₄	—	4	2	no	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄²	#1141	1	1	yes	characteristic
175	C₄²	#1142	1	1	yes	characteristic
176	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
177	C₄²	C₂×C₂²⋊D₇	1	1	yes	characteristic
178	C₁₆	—	28	2	no	—
179	C₂×C₇	—	2	2	no	—
180	C₂×C₇	#1545	1	1	yes	characteristic
181	C₂×C₇	C₂×#358	1	1	yes	characteristic
182	C₂×C₇	C₂×#362	1	1	yes	characteristic
183	C₂³	#6406	1	1	yes	characteristic
184	Q₈	—	4	4	no	—
185	Q₈	—	4	4	no	—
186	Q₈	—	8	4	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	4	4	no	—
191	C₂×C₄	—	8	4	no	—
192	C₂×C₄	#6402	1	2	yes	—
193	C₂×C₄	#6403	1	2	yes	—
194	C₂×C₄	#6406	1	1	yes	characteristic
195	C₂×C₄	#6406	1	1	yes	characteristic
196	C₂×C₄	#6406	1	1	yes	characteristic
197	C₂×C₄	C₂×#1141	1	1	yes	characteristic
198	C₂×C₄	C₂×#1142	1	1	yes	characteristic
199	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
200	C₈	—	2	1	no	—
201	C₈	—	2	1	no	—
202	C₈	—	56	2	no	—
203	C₈	#6407	1	2	yes	—
204	C₈	#6495	1	2	yes	—
205	C₇	#11962	1	1	yes	characteristic, complemented, Sylow
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#89359	1	2	yes	—
209	C₂²	C₂×#6406	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#89359	1	2	yes	—
214	C₄	#89360	1	2	yes	—
215	C₄	C₂×#6402	1	1	yes	characteristic
216	C₄	C₂×#6403	1	1	yes	characteristic
217	C₄	#89987	1	1	yes	characteristic
218	C₄	C₂×#6407	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#11729513	1	1	yes	characteristic
221	C₂	C₂×#89359	1	1	yes	characteristic
222	C₂	C₂×#89360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#6406
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₇
Fitting	C₂×C₇×#494	C₂
socle	C₂²×C₇	C₂×#102
prime radical	C₂³×C₇	#102
cosocle	C₄×C₇×C₈	C₂³
hypercenter	C₂×#494	D₇
hypocenter	C₇	#11962
LCS	G ▷ C₂×C₇×C₈ ▷ C₂×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
4. #11725599	C₂	semidirect
205. C₇	#11962	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
#11727498	2
#11727485	2
#11725599	2
#11962	7