Subgroups — #13271937

Order 1792.

310 conjugacy classes of subgroups.

The subgroup lattice has 196 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#1772	C₂	1	1	yes	maximal, characteristic
2	C₂×#89289	C₂	1	2	yes	maximal
3	#11727518	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	C₂	1	1	yes	maximal, characteristic
5	#11725532	C₂	1	2	yes	maximal
6	C₂×C₇×#108	C₂²	1	1	yes	characteristic
7	C₇×#427	C₂²	1	1	yes	characteristic
8	C₇×#405	—	2	1	no	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6391	—	2	1	no	—
11	C₂×#6391	C₂²	1	2	yes	—
12	C₂×#6386	C₂²	1	2	yes	—
13	#89707	—	2	1	no	—
14	C₇×#339	—	2	1	no	—
15	#89289	—	2	4	no	—
16	#89258	C₄	1	4	yes	complemented
17	#13186	—	7	1	no	maximal, Sylow
18	C₂×C₇×M₁₆	—	2	1	no	—
19	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
20	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
21	C₇×#117	—	2	1	no	—
22	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
23	C₂×C₄⋊Dic₇	—	2	2	no	—
24	C₂×C₄⋊Dic₇	—	4	2	no	—
25	C₇×#108	—	2	1	no	—
26	C₇×#108	D₄	1	2	yes	—
27	C₇×#107	Q₈	1	2	yes	—
28	C₇×#106	—	2	1	no	—
29	C₇×#97	—	4	1	no	—
30	C₄×C₇×C₈	—	4	1	no	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₇×C₈	Q₈	1	2	yes	—
33	#6391	—	2	2	no	—
34	#6391	—	2	2	no	—
35	#6391	—	4	2	no	—
36	#6386	—	2	2	no	—
37	#6386	C₂×C₄	1	4	yes	—
38	C₂×C₇⋊C₁₆	—	2	2	no	—
39	C₂×#365	—	7	2	no	—
40	C₂×#364	—	7	1	no	—
41	#1844	—	7	1	no	—
42	#1772	D₇	1	1	yes	characteristic
43	#1386	—	7	2	no	—
44	C₂²×C₄×C₇	—	2	1	no	—
45	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×Dic₇	—	4	2	no	—
47	C₇×M₁₆	—	2	2	no	—
48	C₇×M₁₆	—	4	1	no	—
49	C₂×C₇×C₈	—	4	1	no	—
50	C₂×C₇×C₈	C₄⋊C₄	1	2	yes	—
51	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
52	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
53	C₂×C₇×C₈	SD₁₆	1	2	yes	—
54	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
56	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
57	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₇×C₈	#41	1	1	yes	characteristic
59	C₂×C₇×C₈	#41	1	1	yes	characteristic
60	C₇×C₄⋊C₄	—	2	2	no	—
61	C₇×C₂²⋊C₄	—	4	1	no	—
62	C₄²×C₇	—	2	1	no	—
63	C₄²×C₇	—	4	1	no	—
64	C₄²×C₇	C₄⋊C₄	1	2	yes	—
65	C₄⋊Dic₇	—	4	2	no	—
66	C₄⋊Dic₇	—	4	2	no	—
67	C₄⋊Dic₇	—	8	4	no	—
68	C₇⋊C₁₆	—	2	4	no	—
69	C₂×M₃₂	—	7	2	no	—
70	C₂×#108	C₂×D₇	1	1	yes	characteristic
71	#427	C₂×D₇	1	1	yes	characteristic
72	C₂×#106	—	7	2	no	—
73	C₂×#106	—	14	1	no	—
74	#405	—	2	1	no	—
75	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
76	#384	—	14	1	no	—
77	#365	—	14	4	no	—
78	#364	—	7	4	no	—
79	#339	—	2	1	no	—
80	C₂³×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
81	C₂×C₄×C₇	—	2	4	no	—
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	4	2	no	—
84	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₇	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₇	#137	1	1	yes	characteristic
90	C₂×Dic₇	—	8	4	no	—
91	C₂×Dic₇	—	8	2	no	—
92	C₇×C₈	—	4	2	no	—
93	C₇×C₈	#106	1	2	yes	—
94	C₇×C₈	#106	1	2	yes	—
95	C₇×C₈	C₄×D₄	1	1	yes	characteristic
96	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
97	C₇×C₈	#125	1	1	yes	characteristic
98	C₇×C₈	C₄⋊Q₈	1	1	yes	characteristic
99	C₂×M₁₆	—	2	1	no	—
100	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
102	#117	—	2	1	no	—
103	C₂×C₄⋊C₄	—	14	2	no	—
104	C₂×C₄⋊C₄	—	28	2	no	—
105	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
106	M₃₂	—	7	4	no	—
107	M₃₂	—	14	2	no	—
108	C₂×C₁₆	—	14	2	no	—
109	#108	—	2	1	no	—
110	#108	D₂₈	1	2	yes	—
111	#107	Dic₁₄	1	2	yes	—
112	#106	—	2	1	no	—
113	#106	—	14	2	no	—
114	#106	—	14	2	no	—
115	#106	—	28	2	no	—
116	#97	—	4	1	no	—
117	C₄×C₈	—	4	1	no	—
118	C₄×C₈	Dic₁₄	1	2	yes	—
119	C₄×C₈	C₄×D₇	1	2	yes	—
120	C₂²×C₇	—	2	2	no	—
121	C₂²×C₇	#428	1	1	yes	characteristic
122	C₂²×C₇	#460	1	1	yes	characteristic
123	C₂²×C₇	#476	1	1	yes	characteristic
124	C₄×C₇	—	2	4	no	—
125	C₄×C₇	—	8	2	no	—
126	C₄×C₇	#389	1	1	yes	characteristic
127	C₄×C₇	C₂×#106	1	1	yes	characteristic
128	C₄×C₇	#461	1	1	yes	characteristic
129	C₄×C₇	#475	1	1	yes	characteristic
130	Dic₇	—	16	2	no	—
131	C₂²×C₄	—	2	1	no	—
132	C₂²×C₄	—	28	2	no	—
133	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
134	M₁₆	—	2	2	no	—
135	M₁₆	—	4	1	no	—
136	C₂×C₈	—	4	1	no	—
137	C₂×C₈	Dic₇⋊C₄	1	2	yes	—
138	C₂×C₈	#1142	1	1	yes	characteristic
139	C₂×C₈	#1142	1	1	yes	characteristic
140	C₂×C₈	#1143	1	2	yes	—
141	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
142	C₂×C₈	C₂×Dic₁₄	1	1	yes	characteristic
143	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
144	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
145	C₂×C₈	#1157	1	1	yes	characteristic
146	C₂×C₈	#1157	1	1	yes	characteristic
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	28	2	no	—
149	C₄⋊C₄	—	28	2	no	—
150	C₄⋊C₄	—	56	4	no	—
151	C₂²⋊C₄	—	4	1	no	—
152	C₄²	—	2	1	no	—
153	C₄²	—	4	1	no	—
154	C₄²	Dic₇⋊C₄	1	2	yes	—
155	C₁₆	—	14	4	no	—
156	C₂×C₇	—	2	1	no	—
157	C₂×C₇	—	2	1	no	—
158	C₂×C₇	#1948	1	1	yes	characteristic
159	C₂×C₇	#2230	1	2	yes	—
160	C₂³	#6512	1	1	yes	characteristic
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	2	no	—
164	C₂×C₄	—	56	4	no	—
165	C₂×C₄	—	56	2	no	—
166	C₂×C₄	C₂×Dic₇⋊C₄	1	1	yes	characteristic
167	C₂×C₄	#6487	1	1	yes	characteristic
168	C₂×C₄	#6493	1	1	yes	characteristic
169	C₂×C₄	C₂×#1142	1	1	yes	characteristic
170	C₂×C₄	C₂×#1143	1	1	yes	characteristic
171	C₂×C₄	#6514	1	1	yes	characteristic
172	C₈	—	4	2	no	—
173	C₈	#6382	1	2	yes	—
174	C₈	#6382	1	2	yes	—
175	C₈	C₄×Dic₁₄	1	1	yes	characteristic
176	C₈	#6432	1	1	yes	characteristic
177	C₈	#6433	1	1	yes	characteristic
178	C₈	C₄×D₂₈	1	1	yes	characteristic
179	C₇	#13186	1	1	yes	characteristic, complemented, Sylow
180	C₂²	—	2	2	no	—
181	C₂²	#89767	1	1	yes	characteristic
182	C₂²	#89780	1	1	yes	characteristic
183	C₂²	#89799	1	1	yes	characteristic
184	C₄	—	2	4	no	—
185	C₄	—	8	2	no	—
186	C₄	—	112	2	no	—
187	C₄	#89700	1	1	yes	characteristic
188	C₄	C₂×#6382	1	1	yes	characteristic
189	C₄	#89791	1	1	yes	characteristic
190	C₄	#89792	1	1	yes	characteristic
191	C₂	—	2	1	no	—
192	C₂	—	2	1	no	—
193	C₂	#11727221	1	1	yes	characteristic
194	C₂	#11728673	1	2	yes	—

	subgroup	G/H
center	C₂²	#89780
commutator	C₂×C₇×C₈	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₇×#1772	C₂
socle	C₂²×C₇	#476
prime radical	C₂³×C₇	AGL(1,ℤ/8)
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	#1772	D₇
hypocenter	C₇	#13186
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
16. #89258	C₄	semidirect
179. C₇	#13186	semidirect

Maximal subgroups

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

subgroup	index
C₇×#1772	2
C₂×#89289	2
#11727518	2
C₂×#89258	2
#11725532	2
#13186	7