Subgroups — C₂×Dic₇×M₃₂ #13506425

Order 1792.

918 conjugacy classes of subgroups.

The subgroup lattice has 252 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×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#6386	C₂	1	1	yes	maximal, characteristic
4	Dic₇×M₃₂	C₂	1	8	yes	maximal, direct product factor
5	C₂×#89297	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₆×Dic₇	C₂	1	2	yes	maximal, complemented
7	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₇	C₄	1	1	yes	characteristic
9	C₂³×C₇⋊C₈	C₄	1	1	yes	characteristic
10	C₂×C₇×M₃₂	C₄	1	8	yes	complemented
11	C₂×C₇×M₃₂	C₂²	1	4	yes	—
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₇	C₄	1	4	yes	—
15	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₇	C₂²	1	4	yes	—
18	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
19	C₂×#6386	C₄	1	8	yes	complemented
20	C₂×#6386	C₂²	1	4	yes	—
21	C₂²×C₇⋊C₁₆	C₂²	1	2	yes	—
22	#89297	C₂²	1	8	yes	complemented
23	C₁₆×Dic₇	C₂²	1	8	yes	complemented
24	C₂×C₄×M₃₂	—	7	1	no	maximal, Sylow
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂³×Dic₇	C₈	1	2	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
31	C₂²×C₇×C₈	C₂³	1	2	yes	—
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₇	C₈	1	4	yes	—
34	C₂×C₄×Dic₇	C₈	1	2	yes	—
35	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	C₈	1	8	yes	—
39	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
41	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
42	C₇×M₃₂	C₂×C₄	1	16	yes	complemented
43	C₂×C₇×C₁₆	C₂×C₄	1	8	yes	complemented
44	C₂×C₇×C₁₆	C₂³	1	4	yes	—
45	C₈×Dic₇	C₂×C₄	1	2	yes	—
46	C₈×Dic₇	C₂×C₄	1	4	yes	—
47	C₈×Dic₇	C₂×C₄	1	4	yes	—
48	C₈×Dic₇	C₂×C₄	1	2	yes	—
49	C₈×Dic₇	C₂³	1	2	yes	—
50	C₈×Dic₇	C₂³	1	2	yes	—
51	#6386	C₂×C₄	1	16	yes	complemented
52	C₂×C₇⋊C₁₆	C₂×C₄	1	8	yes	complemented
53	C₂×C₇⋊C₁₆	C₂³	1	4	yes	—
54	C₂²×M₃₂	—	7	1	no	—
55	C₂²×M₃₂	D₇	1	1	yes	characteristic
56	C₂²×C₄×C₈	—	7	1	no	—
57	C₄×M₃₂	—	7	8	no	—
58	C₂×#346	—	7	2	no	—
59	C₂×C₄×C₁₆	—	7	2	no	—
60	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
61	C₂²×C₄×C₇	C₄²	1	4	yes	—
62	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₇	—	2	8	no	—
68	C₂²×Dic₇	C₂×C₈	1	2	yes	—
69	C₂²×Dic₇	C₂×C₈	1	8	yes	—
70	C₂²×Dic₇	C₂×C₈	1	2	yes	—
71	C₂×C₇×C₈	C₄²	1	8	yes	—
72	C₂×C₇×C₈	C₄²	1	4	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
77	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
79	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
80	C₄×Dic₇	C₂×C₈	1	2	yes	—
81	C₄×Dic₇	C₂×C₈	1	8	yes	—
82	C₄×Dic₇	C₂×C₈	1	2	yes	—
83	C₄×Dic₇	C₂²×C₄	1	2	yes	—
84	C₄×Dic₇	C₂²×C₄	1	2	yes	—
85	C₂×C₇⋊C₈	C₂×C₈	1	16	yes	—
86	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
87	C₂×C₇⋊C₈	C₂×C₈	1	4	yes	—
88	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
89	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
90	C₇×C₁₆	C₂²×C₄	1	8	yes	complemented
91	C₇⋊C₁₆	C₂²×C₄	1	8	yes	complemented
92	C₂³×C₈	—	7	1	no	—
93	C₂³×C₈	C₂×D₇	1	1	yes	characteristic
94	C₂²×C₄²	—	7	1	no	—
95	C₂×M₃₂	—	7	8	no	—
96	C₂×M₃₂	—	7	4	no	—
97	C₂×M₃₂	Dic₇	1	8	yes	direct product factor
98	C₂×M₃₂	C₂×D₇	1	4	yes	—
99	C₂²×C₁₆	—	7	2	no	—
100	C₂²×C₁₆	C₂×D₇	1	2	yes	—
101	C₂×C₄×C₈	—	7	1	no	—
102	C₂×C₄×C₈	—	7	1	no	—
103	C₂×C₄×C₈	—	7	4	no	—
104	C₂×C₄×C₈	—	7	4	no	—
105	C₂×C₄×C₈	—	7	1	no	—
106	C₂×C₄×C₈	—	7	1	no	—
107	#346	—	7	8	no	—
108	C₄×C₁₆	—	7	8	no	—
109	C₂³×C₇	—	2	4	no	—
110	C₂³×C₇	C₄×C₈	1	4	yes	—
111	C₂³×C₇	C₂²×C₈	1	2	yes	—
112	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₇	C₄×C₈	1	8	yes	—
114	C₂×C₄×C₇	C₄×C₈	1	4	yes	—
115	C₂×C₄×C₇	C₂×C₄²	1	4	yes	—
116	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
117	C₂×C₄×C₇	C₂²×C₈	1	4	yes	—
118	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
119	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
120	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
122	C₂×Dic₇	—	2	4	no	—
123	C₂×Dic₇	—	2	16	no	—
124	C₂×Dic₇	M₃₂	1	8	yes	direct product factor
125	C₂×Dic₇	C₂²×C₈	1	4	yes	—
126	C₂×Dic₇	C₂²×C₈	1	4	yes	—
127	C₇×C₈	C₂×C₄²	1	2	yes	—
128	C₇×C₈	C₂×C₄²	1	4	yes	—
129	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
130	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
131	C₇⋊C₈	C₂²×C₈	1	4	yes	—
132	C₇⋊C₈	C₂²×C₈	1	4	yes	—
133	C₂³×C₄	—	7	2	no	—
134	C₂³×C₄	C₄×D₇	1	1	yes	characteristic
135	C₂²×C₈	—	7	8	no	—
136	C₂²×C₈	—	7	4	no	—
137	C₂²×C₈	—	7	1	no	—
138	C₂²×C₈	—	7	1	no	—
139	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
140	C₂²×C₈	C₄×D₇	1	2	yes	—
141	C₂²×C₈	C₄×D₇	1	4	yes	—
142	C₂²×C₈	C₂×Dic₇	1	4	yes	—
143	C₂²×C₈	C₂²×D₇	1	2	yes	—
144	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
145	C₂×C₄²	—	7	2	no	—
146	C₂×C₄²	—	7	4	no	—
147	C₂×C₄²	—	7	1	no	—
148	C₂×C₄²	—	7	4	no	—
149	C₂×C₄²	—	7	1	no	—
150	M₃₂	—	7	16	no	—
151	M₃₂	C₂×Dic₇	1	16	yes	direct product factor
152	C₂×C₁₆	—	7	8	no	—
153	C₂×C₁₆	—	7	4	no	—
154	C₂×C₁₆	C₂×Dic₇	1	8	yes	complemented
155	C₂×C₁₆	C₂²×D₇	1	4	yes	—
156	C₄×C₈	—	7	4	no	—
157	C₄×C₈	—	7	4	no	—
158	C₄×C₈	—	7	2	no	—
159	C₄×C₈	—	7	2	no	—
160	C₄×C₈	—	7	2	no	—
161	C₄×C₈	—	7	2	no	—
162	C₂²×C₇	—	2	8	no	—
163	C₂²×C₇	—	2	4	no	—
164	C₂²×C₇	C₂×C₄×C₈	1	4	yes	—
165	C₂²×C₇	C₂×C₄×C₈	1	2	yes	—
166	C₂²×C₇	C₂×M₃₂	1	4	yes	—
167	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
168	C₄×C₇	C₂×C₄×C₈	1	2	yes	—
169	C₄×C₇	C₂×C₄×C₈	1	4	yes	—
170	C₄×C₇	C₂²×C₄²	1	1	yes	characteristic
171	C₄×C₇	C₂³×C₈	1	1	yes	characteristic
172	Dic₇	—	2	4	no	—
173	Dic₇	C₂×M₃₂	1	8	yes	direct product factor
174	C₂⁴	C₈×D₇	1	1	yes	characteristic
175	C₂²×C₄	—	7	2	no	—
176	C₂²×C₄	—	7	8	no	—
177	C₂²×C₄	—	7	2	no	—
178	C₂²×C₄	—	14	8	no	—
179	C₂²×C₄	C₈×D₇	1	4	yes	—
180	C₂²×C₄	C₈×D₇	1	2	yes	—
181	C₂²×C₄	C₈×D₇	1	1	yes	characteristic
182	C₂²×C₄	C₄×Dic₇	1	4	yes	—
183	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
184	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
185	C₂×C₈	—	7	16	no	—
186	C₂×C₈	—	7	4	no	—
187	C₂×C₈	—	7	4	no	—
188	C₂×C₈	—	7	2	no	—
189	C₂×C₈	—	7	2	no	—
190	C₂×C₈	C₄×Dic₇	1	4	yes	—
191	C₂×C₈	C₄×Dic₇	1	8	yes	—
192	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
193	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
194	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
195	C₂×C₈	C₂×C₄×D₇	1	4	yes	—
196	C₂×C₈	C₂²×Dic₇	1	2	yes	—
197	C₂×C₈	C₂²×Dic₇	1	4	yes	—
198	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
199	C₄²	—	7	2	no	—
200	C₄²	—	7	8	no	—
201	C₄²	—	7	2	no	—
202	C₄²	—	7	2	no	—
203	C₄²	—	7	2	no	—
204	C₁₆	—	7	8	no	—
205	C₁₆	C₂²×Dic₇	1	8	yes	complemented
206	C₂×C₇	—	2	4	no	—
207	C₂×C₇	C₄×M₃₂	1	4	yes	complemented
208	C₂×C₇	C₂²×C₄×C₈	1	1	yes	characteristic
209	C₂×C₇	C₂²×M₃₂	1	2	yes	—
210	C₂³	—	2	4	no	—
211	C₂³	C₈×Dic₇	1	4	yes	—
212	C₂³	C₂×C₈×D₇	1	2	yes	—
213	C₂³	C₂×C₈×D₇	1	1	yes	characteristic
214	C₂×C₄	—	7	4	no	—
215	C₂×C₄	—	7	8	no	—
216	C₂×C₄	—	7	4	no	—
217	C₂×C₄	—	14	16	no	—
218	C₂×C₄	—	14	4	no	—
219	C₂×C₄	C₈×Dic₇	1	8	yes	—
220	C₂×C₄	C₈×Dic₇	1	4	yes	—
221	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
222	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
223	C₂×C₄	C₂×C₈×D₇	1	4	yes	—
224	C₂×C₄	C₂×C₈×D₇	1	2	yes	—
225	C₂×C₄	C₂×C₄×Dic₇	1	4	yes	—
226	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
227	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
228	C₈	—	7	4	no	—
229	C₈	—	7	4	no	—
230	C₈	C₂×C₄×Dic₇	1	4	yes	—
231	C₈	C₂×C₄×Dic₇	1	2	yes	—
232	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
233	C₈	C₂³×Dic₇	1	1	yes	characteristic
234	C₇	C₂×C₄×M₃₂	1	1	yes	characteristic, complemented, Sylow
235	C₂²	—	2	8	no	—
236	C₂²	—	2	4	no	—
237	C₂²	D₇×M₃₂	1	4	yes	—
238	C₂²	C₂×C₈×Dic₇	1	4	yes	—
239	C₂²	C₂×C₈×Dic₇	1	2	yes	—
240	C₂²	C₂²×C₈×D₇	1	1	yes	characteristic
241	C₄	—	7	8	no	—
242	C₄	—	14	4	no	—
243	C₄	C₂×C₈×Dic₇	1	4	yes	—
244	C₄	C₂×C₈×Dic₇	1	2	yes	—
245	C₄	C₂²×C₈×D₇	1	1	yes	characteristic
246	C₄	C₂²×C₄×Dic₇	1	1	yes	characteristic
247	C₂	—	2	4	no	—
248	C₂	Dic₇×M₃₂	1	4	yes	direct product factor
249	C₂	C₂×D₇×M₃₂	1	2	yes	—
250	C₂	C₂²×C₈×Dic₇	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
4. Dic₇×M₃₂	C₂	direct (N × H)
5. C₂×#89297	C₂	semidirect
6. C₂×C₁₆×Dic₇	C₂	semidirect
10. C₂×C₇×M₃₂	C₄	semidirect
19. C₂×#6386	C₄	semidirect
22. #89297	C₂²	semidirect
23. C₁₆×Dic₇	C₂²	semidirect
42. C₇×M₃₂	C₂×C₄	semidirect
43. C₂×C₇×C₁₆	C₂×C₄	semidirect
51. #6386	C₂×C₄	semidirect
52. C₂×C₇⋊C₁₆	C₂×C₄	semidirect
90. C₇×C₁₆	C₂²×C₄	semidirect
91. C₇⋊C₁₆	C₂²×C₄	semidirect
97. C₂×M₃₂	Dic₇	direct (N × H)
124. C₂×Dic₇	M₃₂	direct (N × H)
151. M₃₂	C₂×Dic₇	direct (N × H)
154. C₂×C₁₆	C₂×Dic₇	semidirect
173. Dic₇	C₂×M₃₂	direct (N × H)
205. C₁₆	C₂²×Dic₇	semidirect
207. C₂×C₇	C₄×M₃₂	semidirect
234. C₇	C₂×C₄×M₃₂	semidirect
248. C₂	Dic₇×M₃₂	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×C₈×Dic₇	2
C₂²×#6386	2
Dic₇×M₃₂	2
C₂×#89297	2
C₂×C₁₆×Dic₇	2
C₂×C₄×M₃₂	7

Subgroups — C2×Dic7×M32 #13506425

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×Dic₇×M₃₂ #13506425