Subgroups — C₂²×#89307 #13506449

Order 1792.

782 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6389	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	C₂×#89307	C₂	1	12	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
6	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
7	C₂³×C₇⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	C₄	1	6	yes	complemented
9	C₂×C₇×M₃₂	C₂²	1	6	yes	—
10	C₂²×C₇×C₁₆	C₄	1	2	yes	complemented
11	C₂×#6389	C₄	1	6	yes	—
12	C₂×#6389	C₂²	1	6	yes	—
13	C₂×#6386	C₄	1	6	yes	complemented
14	C₂×#6386	C₂²	1	6	yes	—
15	C₂²×C₇⋊C₁₆	C₄	1	2	yes	complemented
16	#89307	C₂²	1	16	yes	direct product factor
17	C₂²×#347	—	7	1	no	maximal, Sylow
18	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
19	C₂³×Dic₇	—	2	1	no	—
20	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
23	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₇×C₈	C₂×C₄	1	3	yes	—
25	C₂²×C₇×C₈	C₂³	1	3	yes	—
26	C₂×C₄×Dic₇	—	2	3	no	—
27	C₂×C₄×Dic₇	C₂×C₄	1	6	yes	—
28	C₂²×C₇⋊C₈	—	2	3	no	—
29	C₂²×C₇⋊C₈	—	2	1	no	—
30	C₂²×C₇⋊C₈	C₂×C₄	1	6	yes	—
31	C₇×M₃₂	C₂×C₄	1	12	yes	complemented
32	C₇×M₃₂	C₂³	1	4	yes	—
33	C₂×C₇×C₁₆	C₂×C₄	1	12	yes	complemented
34	#6389	C₂×C₄	1	12	yes	—
35	#6389	C₂³	1	4	yes	—
36	#6386	C₂×C₄	1	12	yes	complemented
37	#6386	C₂³	1	4	yes	—
38	C₂×C₇⋊C₁₆	C₂×C₄	1	12	yes	complemented
39	C₂²×M₃₂	—	7	1	no	—
40	C₂²×M₃₂	D₇	1	1	yes	characteristic
41	C₂²×#97	—	7	1	no	—
42	C₂×#347	—	7	12	no	—
43	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
44	C₂²×C₄×C₇	C₄²	1	3	yes	—
45	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
46	C₂²×C₄×C₇	M₁₆	1	3	yes	—
47	C₂²×C₄×C₇	M₁₆	1	3	yes	—
48	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
50	C₂²×Dic₇	—	2	6	no	—
51	C₂²×Dic₇	—	4	4	no	—
52	C₂×C₇×C₈	C₄²	1	3	yes	—
53	C₂×C₇×C₈	C₄²	1	3	yes	—
54	C₂×C₇×C₈	C₄²	1	6	yes	—
55	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
58	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
59	C₂×C₇×C₈	C₂²×C₄	1	3	yes	—
60	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
61	C₄×Dic₇	—	2	6	no	—
62	C₄×Dic₇	C₂²×C₄	1	4	yes	—
63	C₂×C₇⋊C₈	—	2	6	no	—
64	C₂×C₇⋊C₈	—	2	6	no	—
65	C₂×C₇⋊C₈	C₂²×C₄	1	4	yes	—
66	C₇×C₁₆	C₂²×C₄	1	8	yes	complemented
67	C₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
68	C₂³×C₈	—	7	1	no	—
69	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
70	C₂²×C₄²	—	7	1	no	—
71	C₂×M₃₂	—	7	6	no	—
72	C₂×M₃₂	—	7	6	no	—
73	C₂×M₃₂	Dic₇	1	6	yes	complemented
74	C₂×M₃₂	C₂×D₇	1	6	yes	—
75	C₂²×C₁₆	—	7	2	no	—
76	C₂²×C₁₆	Dic₇	1	2	yes	complemented
77	C₂×#97	—	7	6	no	—
78	C₂×#97	—	7	6	no	—
79	#347	—	7	16	no	—
80	C₂³×C₇	—	2	4	no	—
81	C₂³×C₇	#97	1	3	yes	—
82	C₂³×C₇	#97	1	1	yes	characteristic
83	C₂³×C₇	C₂×M₁₆	1	3	yes	—
84	C₂×C₄×C₇	#97	1	6	yes	—
85	C₂×C₄×C₇	#97	1	3	yes	—
86	C₂×C₄×C₇	#97	1	3	yes	—
87	C₂×C₄×C₇	C₂×C₄²	1	3	yes	—
88	C₂×C₄×C₇	C₂×C₄²	1	3	yes	—
89	C₂×C₄×C₇	C₂×M₁₆	1	3	yes	—
90	C₂×C₄×C₇	C₂×M₁₆	1	3	yes	—
91	C₂×C₄×C₇	C₂×M₁₆	1	3	yes	—
92	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
93	C₂×Dic₇	—	2	4	no	—
94	C₂×Dic₇	—	4	12	no	—
95	C₇×C₈	C₂×C₄²	1	3	yes	—
96	C₇×C₈	C₂×C₄²	1	3	yes	—
97	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
98	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
99	C₇⋊C₈	—	2	4	no	—
100	C₂³×C₄	—	14	1	no	—
101	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
102	C₂²×C₈	—	7	6	no	—
103	C₂²×C₈	—	14	1	no	—
104	C₂²×C₈	—	14	3	no	—
105	C₂²×C₈	C₄×D₇	1	3	yes	—
106	C₂²×C₈	C₄×D₇	1	3	yes	—
107	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
108	C₂²×C₈	C₂×Dic₇	1	3	yes	—
109	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
110	C₂²×C₈	C₂²×D₇	1	3	yes	—
111	C₂×C₄²	—	7	6	no	—
112	C₂×C₄²	—	14	3	no	—
113	M₃₂	—	7	12	no	—
114	M₃₂	—	7	4	no	—
115	M₃₂	C₂×Dic₇	1	12	yes	complemented
116	M₃₂	C₂²×D₇	1	4	yes	—
117	C₂×C₁₆	—	7	12	no	—
118	C₂×C₁₆	C₂×Dic₇	1	12	yes	complemented
119	#97	—	7	12	no	—
120	#97	—	7	4	no	—
121	C₂²×C₇	—	2	6	no	—
122	C₂²×C₇	—	2	6	no	—
123	C₂²×C₇	#347	1	4	yes	complemented
124	C₂²×C₇	C₂×#97	1	3	yes	—
125	C₂²×C₇	C₂×#97	1	3	yes	—
126	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
127	C₄×C₇	C₂×#97	1	3	yes	—
128	C₄×C₇	C₂×#97	1	3	yes	—
129	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
130	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
131	Dic₇	—	4	4	no	—
132	C₂⁴	#1131	1	1	yes	characteristic
133	C₂²×C₄	—	14	6	no	—
134	C₂²×C₄	—	28	4	no	—
135	C₂²×C₄	#1131	1	1	yes	characteristic
136	C₂²×C₄	#1131	1	3	yes	—
137	C₂²×C₄	#1131	1	3	yes	—
138	C₂²×C₄	C₄×Dic₇	1	3	yes	—
139	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
140	C₂²×C₄	C₂×C₄×D₇	1	3	yes	—
141	C₂×C₈	—	7	4	no	—
142	C₂×C₈	—	14	6	no	—
143	C₂×C₈	—	14	6	no	—
144	C₂×C₈	C₄×Dic₇	1	6	yes	—
145	C₂×C₈	C₄×Dic₇	1	3	yes	—
146	C₂×C₈	C₄×Dic₇	1	3	yes	—
147	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
148	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
149	C₂×C₈	C₂×C₄×D₇	1	3	yes	—
150	C₂×C₈	C₂²×Dic₇	1	3	yes	—
151	C₂×C₈	C₂²×Dic₇	1	3	yes	—
152	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
153	C₄²	—	7	4	no	—
154	C₄²	—	14	6	no	—
155	C₁₆	—	7	8	no	—
156	C₁₆	C₂²×Dic₇	1	8	yes	complemented
157	C₂×C₇	—	2	1	no	—
158	C₂×C₇	—	2	3	no	—
159	C₂×C₇	C₂×#347	1	6	yes	complemented
160	C₂×C₇	C₂²×#97	1	1	yes	characteristic
161	C₂³	—	2	4	no	—
162	C₂³	#6389	1	3	yes	—
163	C₂³	#6389	1	1	yes	characteristic
164	C₂³	C₂×#1131	1	3	yes	—
165	C₂×C₄	—	14	4	no	—
166	C₂×C₄	—	28	12	no	—
167	C₂×C₄	#6389	1	6	yes	—
168	C₂×C₄	#6389	1	3	yes	—
169	C₂×C₄	#6389	1	3	yes	—
170	C₂×C₄	C₂×#1131	1	3	yes	—
171	C₂×C₄	C₂×#1131	1	3	yes	—
172	C₂×C₄	C₂×#1131	1	3	yes	—
173	C₂×C₄	C₂×C₄×Dic₇	1	3	yes	—
174	C₂×C₄	C₂×C₄×Dic₇	1	3	yes	—
175	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
176	C₈	—	14	4	no	—
177	C₈	C₂×C₄×Dic₇	1	3	yes	—
178	C₈	C₂×C₄×Dic₇	1	3	yes	—
179	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
180	C₈	C₂³×Dic₇	1	1	yes	characteristic
181	C₇	C₂²×#347	1	1	yes	characteristic, complemented, Sylow
182	C₂²	—	2	6	no	—
183	C₂²	—	2	6	no	—
184	C₂²	#89307	1	4	yes	direct product factor
185	C₂²	C₂×#6389	1	3	yes	—
186	C₂²	C₂×#6389	1	3	yes	—
187	C₂²	C₂²×#1131	1	1	yes	characteristic
188	C₄	—	28	4	no	—
189	C₄	C₂×#6389	1	3	yes	—
190	C₄	C₂×#6389	1	3	yes	—
191	C₄	C₂²×#1131	1	1	yes	characteristic
192	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
193	C₂	—	2	1	no	—
194	C₂	—	2	3	no	—
195	C₂	C₂×#89307	1	6	yes	direct product factor
196	C₂	C₂²×#6389	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄×Dic₇
commutator	C₄×C₇	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂²×C₇×M₃₂	C₂
socle	C₂³×C₇	#97
prime radical	C₂⁴×C₇	M₁₆
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂²×M₃₂	D₇
hypocenter	C₇	C₂²×#347
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
4. C₂×#89307	C₂	direct (N × H)
8. C₂×C₇×M₃₂	C₄	semidirect
10. C₂²×C₇×C₁₆	C₄	semidirect
13. C₂×#6386	C₄	semidirect
15. C₂²×C₇⋊C₁₆	C₄	semidirect
16. #89307	C₂²	direct (N × H)
31. C₇×M₃₂	C₂×C₄	semidirect
33. C₂×C₇×C₁₆	C₂×C₄	semidirect
36. #6386	C₂×C₄	semidirect
38. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
66. C₇×C₁₆	C₂²×C₄	semidirect
67. C₇⋊C₁₆	C₂²×C₄	semidirect
73. C₂×M₃₂	Dic₇	semidirect
76. C₂²×C₁₆	Dic₇	semidirect
115. M₃₂	C₂×Dic₇	semidirect
118. C₂×C₁₆	C₂×Dic₇	semidirect
123. C₂²×C₇	#347	semidirect
156. C₁₆	C₂²×Dic₇	semidirect
159. C₂×C₇	C₂×#347	semidirect
181. C₇	C₂²×#347	semidirect
184. C₂²	#89307	direct (N × H)
195. C₂	C₂×#89307	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₇×M₃₂	2
C₂²×#6389	2
C₂²×#6386	2
C₂×#89307	2
C₂²×#347	7

Subgroups — C22×#89307 #13506449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#89307 #13506449