Subgroups — C₂×C₇×#932 #13061515

Order 1344.

596 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₇×D₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₇×#235	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#214	C₂	1	2	yes	maximal, complemented
4	C₇×#932	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₇×#213	C₂	1	2	yes	maximal, complemented
6	C₂×C₇×#208	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₇×AGL(1,ℤ/8)	—	3	1	no	maximal, Hall
8	C₂⁴×C₃×C₇	—	2	1	no	—
9	C₂×C₃×C₇×D₄	—	2	1	no	—
10	C₂×C₃×C₇×D₄	—	2	2	no	—
11	C₂×C₃×C₇×D₄	C₂²	1	2	yes	—
12	C₂×C₃×C₇×D₄	C₂²	1	4	yes	complemented
13	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
14	C₂×C₇×#67	—	2	1	no	—
15	C₇×#235	—	2	2	no	—
16	C₇×#235	C₂²	1	4	yes	complemented
17	C₂×C₇×D₁₂	C₂²	1	1	yes	characteristic, complemented
18	C₂×C₄×D₃×C₇	—	2	1	no	—
19	C₂×C₇×Dic₆	C₂²	1	1	yes	characteristic, complemented
20	C₇×#214	C₂²	1	8	yes	complemented
21	C₇×#213	C₂²	1	8	yes	complemented
22	C₇×#208	C₂²	1	4	yes	complemented
23	C₂×C₇×C₃⋊C₈	C₂²	1	2	yes	complemented
24	C₂×C₇×#41	—	3	1	no	—
25	C₂²×C₇×D₄	D₃	1	1	yes	characteristic, complemented
26	C₇×AGL(1,ℤ/8)	—	3	8	no	—
27	C₂×C₇×SD₁₆	—	3	2	no	—
28	C₂×C₇×D₈	—	3	2	no	—
29	C₂×C₇×M₁₆	—	3	1	no	—
30	C₂×#932	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
31	C₂³×C₃×C₇	—	2	2	no	—
32	C₂³×C₃×C₇	—	2	4	no	—
33	C₂³×C₃×C₇	—	4	4	no	—
34	C₂³×C₃×C₇	D₄	1	1	yes	characteristic
35	C₂²×D₃×C₇	—	2	1	no	—
36	C₃×C₇×D₄	—	2	2	no	—
37	C₃×C₇×D₄	—	2	4	no	—
38	C₃×C₇×D₄	C₂³	1	4	yes	—
39	C₂×C₃×C₄×C₇	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₇	D₄	1	2	yes	—
41	C₂×C₃×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
43	C₇×#67	—	2	4	no	—
44	C₂×C₇×Dic₃	—	2	1	no	—
45	C₇×D₁₂	—	2	1	no	—
46	C₇×D₁₂	C₂³	1	2	yes	complemented
47	C₄×D₃×C₇	—	2	4	no	—
48	C₇×Dic₆	—	2	1	no	—
49	C₇×Dic₆	C₂³	1	2	yes	complemented
50	C₇×C₃⋊C₈	C₂³	1	4	yes	complemented
51	C₂⁴×C₇	—	2	1	no	—
52	C₇×#41	—	3	4	no	—
53	C₇×#41	—	6	2	no	—
54	C₂×C₇×Q₈	—	3	1	no	—
55	C₂×C₇×D₄	—	2	2	no	—
56	C₂×C₇×D₄	—	2	1	no	—
57	C₂×C₇×D₄	—	3	1	no	—
58	C₂×C₇×D₄	—	6	1	no	—
59	C₂×C₇×D₄	C₂×D₃	1	2	yes	—
60	C₂×C₇×D₄	C₂×D₃	1	4	yes	complemented
61	C₂²×C₄×C₇	—	6	1	no	—
62	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
63	C₇×SD₁₆	—	3	8	no	—
64	C₇×D₈	—	3	8	no	—
65	C₇×M₁₆	—	3	4	no	—
66	C₂×C₇×C₈	—	3	2	no	—
67	C₂²×C₃×D₄	C₂×C₇	1	1	yes	characteristic, complemented
68	C₂×#235	C₂×C₇	1	1	yes	characteristic, complemented
69	C₂×#214	C₂×C₇	1	2	yes	complemented
70	#932	C₂×C₇	1	8	yes	direct product factor
71	C₂×#213	C₂×C₇	1	2	yes	complemented
72	C₂×#208	C₂×C₇	1	1	yes	characteristic, complemented
73	C₂²×C₃×C₇	—	2	2	no	—
74	C₂²×C₃×C₇	—	2	4	no	—
75	C₂²×C₃×C₇	—	4	8	no	—
76	C₂²×C₃×C₇	—	4	4	no	—
77	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
78	C₂²×C₃×C₇	C₂×D₄	1	1	yes	characteristic
79	C₂×D₃×C₇	—	2	2	no	—
80	C₂×D₃×C₇	—	4	2	no	—
81	C₃×C₄×C₇	C₂×D₄	1	1	yes	characteristic
82	C₃×C₄×C₇	C₂×D₄	1	2	yes	—
83	C₃×C₄×C₇	C₂⁴	1	1	yes	characteristic
84	C₇×Dic₃	—	2	2	no	—
85	C₂×AGL(1,ℤ/8)	—	3	1	no	Sylow
86	C₂³×C₇	—	2	4	no	—
87	C₂³×C₇	—	2	2	no	—
88	C₂³×C₇	—	4	4	no	—
89	C₂³×C₇	—	6	1	no	—
90	C₂³×C₇	#67	1	1	yes	characteristic
91	C₇×Q₈	—	3	2	no	—
92	C₇×Q₈	—	6	1	no	—
93	C₇×D₄	—	2	2	no	—
94	C₇×D₄	—	2	4	no	—
95	C₇×D₄	—	3	2	no	—
96	C₇×D₄	—	6	4	no	—
97	C₇×D₄	—	6	1	no	—
98	C₇×D₄	C₂²×D₃	1	4	yes	—
99	C₂×C₄×C₇	—	6	1	no	—
100	C₂×C₄×C₇	—	6	4	no	—
101	C₂×C₄×C₇	#67	1	2	yes	—
102	C₂×C₄×C₇	#67	1	1	yes	characteristic
103	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
104	C₂×C₄×C₇	C₂²×D₃	1	1	yes	characteristic
105	C₇×C₈	—	3	4	no	—
106	C₂⁴×C₃	—	2	1	no	—
107	C₂×C₃×D₄	—	2	1	no	—
108	C₂×C₃×D₄	—	2	2	no	—
109	C₂×C₃×D₄	C₂²×C₇	1	2	yes	—
110	C₂×C₃×D₄	C₂²×C₇	1	4	yes	complemented
111	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
112	C₂×#67	—	2	1	no	—
113	#235	—	2	2	no	—
114	#235	C₂²×C₇	1	4	yes	complemented
115	C₂×D₁₂	C₂²×C₇	1	1	yes	characteristic, complemented
116	C₂×C₄×D₃	—	2	1	no	—
117	C₂×Dic₆	C₂²×C₇	1	1	yes	characteristic, complemented
118	#214	C₂²×C₇	1	8	yes	complemented
119	#213	C₂²×C₇	1	8	yes	complemented
120	#208	C₂²×C₇	1	4	yes	complemented
121	C₂×C₃⋊C₈	C₂²×C₇	1	2	yes	complemented
122	C₂×C₃×C₇	—	2	2	no	—
123	C₂×C₃×C₇	—	4	4	no	—
124	C₂×C₃×C₇	AGL(1,ℤ/8)	1	2	yes	complemented
125	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
126	D₃×C₇	—	4	2	no	—
127	C₂×#41	—	3	1	no	—
128	C₂²×D₄	D₃×C₇	1	1	yes	characteristic, complemented
129	AGL(1,ℤ/8)	—	3	8	no	—
130	C₂×SD₁₆	—	3	2	no	—
131	C₂×D₈	—	3	2	no	—
132	C₂×M₁₆	—	3	1	no	—
133	C₂²×C₇	—	2	2	no	—
134	C₂²×C₇	—	2	4	no	—
135	C₂²×C₇	—	4	8	no	—
136	C₂²×C₇	—	4	4	no	—
137	C₂²×C₇	—	6	2	no	—
138	C₂²×C₇	—	12	2	no	—
139	C₂²×C₇	C₂×#67	1	2	yes	—
140	C₂²×C₇	C₂×#67	1	1	yes	characteristic
141	C₄×C₇	—	6	2	no	—
142	C₄×C₇	C₂×#67	1	2	yes	—
143	C₄×C₇	C₂×#67	1	1	yes	characteristic
144	C₄×C₇	C₂³×D₃	1	1	yes	characteristic
145	C₂³×C₃	—	2	2	no	—
146	C₂³×C₃	—	2	4	no	—
147	C₂³×C₃	—	4	4	no	—
148	C₂³×C₃	C₇×D₄	1	1	yes	characteristic
149	C₂²×D₃	—	2	1	no	—
150	C₃×D₄	—	2	2	no	—
151	C₃×D₄	—	2	4	no	—
152	C₃×D₄	C₂³×C₇	1	4	yes	—
153	C₂×C₃×C₄	C₇×D₄	1	1	yes	characteristic
154	C₂×C₃×C₄	C₇×D₄	1	2	yes	—
155	C₂×C₃×C₄	C₂³×C₇	1	1	yes	characteristic
156	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
157	#67	—	2	4	no	—
158	C₂×Dic₃	—	2	1	no	—
159	D₁₂	—	2	1	no	—
160	D₁₂	C₂³×C₇	1	2	yes	complemented
161	C₄×D₃	—	2	4	no	—
162	Dic₆	—	2	1	no	—
163	Dic₆	C₂³×C₇	1	2	yes	complemented
164	C₃⋊C₈	C₂³×C₇	1	4	yes	complemented
165	C₃×C₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Hall
166	C₂⁴	—	2	1	no	—
167	#41	—	3	4	no	—
168	#41	—	6	2	no	—
169	C₂×Q₈	—	3	1	no	—
170	C₂×D₄	—	2	1	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	3	1	no	—
173	C₂×D₄	—	6	1	no	—
174	C₂×D₄	C₂×D₃×C₇	1	2	yes	—
175	C₂×D₄	C₂×D₃×C₇	1	4	yes	complemented
176	C₂²×C₄	—	6	1	no	—
177	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
178	SD₁₆	—	3	8	no	—
179	D₈	—	3	8	no	—
180	M₁₆	—	3	4	no	—
181	C₂×C₈	—	3	2	no	—
182	C₂×C₇	—	2	2	no	—
183	C₂×C₇	—	4	4	no	—
184	C₂×C₇	—	12	2	no	—
185	C₂×C₇	#932	1	2	yes	direct product factor
186	C₂×C₇	C₂²×#67	1	1	yes	characteristic
187	C₂²×C₃	—	2	2	no	—
188	C₂²×C₃	—	2	4	no	—
189	C₂²×C₃	—	4	8	no	—
190	C₂²×C₃	—	4	4	no	—
191	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
192	C₂²×C₃	C₂×C₇×D₄	1	1	yes	characteristic
193	C₂×D₃	—	2	2	no	—
194	C₂×D₃	—	4	2	no	—
195	C₃×C₄	C₂×C₇×D₄	1	1	yes	characteristic
196	C₃×C₄	C₂×C₇×D₄	1	2	yes	—
197	C₃×C₄	C₂⁴×C₇	1	1	yes	characteristic
198	Dic₃	—	2	2	no	—
199	C₂³	—	2	2	no	—
200	C₂³	—	2	4	no	—
201	C₂³	—	4	4	no	—
202	C₂³	—	6	1	no	—
203	C₂³	C₇×#67	1	1	yes	characteristic
204	Q₈	—	3	2	no	—
205	Q₈	—	6	1	no	—
206	D₄	—	2	2	no	—
207	D₄	—	2	4	no	—
208	D₄	—	3	2	no	—
209	D₄	—	6	4	no	—
210	D₄	—	6	1	no	—
211	D₄	C₂²×D₃×C₇	1	4	yes	—
212	C₂×C₄	—	6	1	no	—
213	C₂×C₄	—	6	4	no	—
214	C₂×C₄	C₇×#67	1	1	yes	characteristic
215	C₂×C₄	C₇×#67	1	2	yes	—
216	C₂×C₄	C₂²×D₃×C₇	1	1	yes	characteristic
217	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
218	C₈	—	3	4	no	—
219	C₇	C₂×#932	1	1	yes	characteristic, direct product factor, Sylow
220	C₂×C₃	—	2	2	no	—
221	C₂×C₃	—	4	4	no	—
222	C₂×C₃	C₇×AGL(1,ℤ/8)	1	2	yes	complemented
223	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
224	D₃	—	4	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	4	8	no	—
228	C₂²	—	4	4	no	—
229	C₂²	—	6	2	no	—
230	C₂²	—	12	2	no	—
231	C₂²	C₂×C₇×#67	1	2	yes	—
232	C₂²	C₂×C₇×#67	1	1	yes	characteristic
233	C₄	—	6	2	no	—
234	C₄	C₂×C₇×#67	1	2	yes	—
235	C₄	C₂×C₇×#67	1	1	yes	characteristic
236	C₄	C₂³×D₃×C₇	1	1	yes	characteristic
237	C₃	C₂×C₇×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
238	C₂	—	2	2	no	—
239	C₂	—	4	4	no	—
240	C₂	—	12	2	no	—
241	C₂	C₇×#932	1	2	yes	direct product factor
242	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₇	C₂×#67
commutator, cosocle	C₃×C₄	C₂⁴×C₇
Frattini	C₄	C₂³×D₃×C₇
Fitting	C₂²×C₃×C₇×D₄	C₂
socle	C₂²×C₃×C₇	C₂×D₄
hypercenter	C₂²×C₇×D₄	D₃
hypocenter	C₃	C₂×C₇×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₇ ◁ C₂²×C₄×C₇ ◁ C₂²×C₇×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₇×D₄	C₂	semidirect
2. C₂×C₇×#235	C₂	semidirect
3. C₂×C₇×#214	C₂	semidirect
4. C₇×#932	C₂	direct (N × H)
5. C₂×C₇×#213	C₂	semidirect
6. C₂×C₇×#208	C₂	semidirect
12. C₂×C₃×C₇×D₄	C₂²	semidirect
16. C₇×#235	C₂²	semidirect
17. C₂×C₇×D₁₂	C₂²	semidirect
19. C₂×C₇×Dic₆	C₂²	semidirect
20. C₇×#214	C₂²	semidirect
21. C₇×#213	C₂²	semidirect
22. C₇×#208	C₂²	semidirect
23. C₂×C₇×C₃⋊C₈	C₂²	semidirect
25. C₂²×C₇×D₄	D₃	semidirect
30. C₂×#932	C₇	direct (N × H)
46. C₇×D₁₂	C₂³	semidirect
49. C₇×Dic₆	C₂³	semidirect
50. C₇×C₃⋊C₈	C₂³	semidirect
60. C₂×C₇×D₄	C₂×D₃	semidirect
67. C₂²×C₃×D₄	C₂×C₇	semidirect
68. C₂×#235	C₂×C₇	semidirect
69. C₂×#214	C₂×C₇	semidirect
70. #932	C₂×C₇	direct (N × H)
71. C₂×#213	C₂×C₇	semidirect
72. C₂×#208	C₂×C₇	semidirect
110. C₂×C₃×D₄	C₂²×C₇	semidirect
114. #235	C₂²×C₇	semidirect
115. C₂×D₁₂	C₂²×C₇	semidirect
117. C₂×Dic₆	C₂²×C₇	semidirect
118. #214	C₂²×C₇	semidirect
119. #213	C₂²×C₇	semidirect
120. #208	C₂²×C₇	semidirect
121. C₂×C₃⋊C₈	C₂²×C₇	semidirect
124. C₂×C₃×C₇	AGL(1,ℤ/8)	semidirect
128. C₂²×D₄	D₃×C₇	semidirect
160. D₁₂	C₂³×C₇	semidirect
163. Dic₆	C₂³×C₇	semidirect
164. C₃⋊C₈	C₂³×C₇	semidirect
165. C₃×C₇	C₂×AGL(1,ℤ/8)	semidirect
175. C₂×D₄	C₂×D₃×C₇	semidirect
185. C₂×C₇	#932	direct (N × H)
219. C₇	C₂×#932	direct (N × H)
222. C₂×C₃	C₇×AGL(1,ℤ/8)	semidirect
237. C₃	C₂×C₇×AGL(1,ℤ/8)	semidirect
241. C₂	C₇×#932	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×C₃×C₇×D₄	2
C₂×C₇×#235	2
C₂×C₇×#214	2
C₇×#932	2
C₂×C₇×#213	2
C₂×C₇×#208	2
C₂×C₇×AGL(1,ℤ/8)	3
C₂×#932	7

Subgroups — C2×C7×#932 #13061515

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#932 #13061515