Subgroups — C₇×#12732 #13189948

Order 1792.

934 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#358	C₂	1	4	yes	maximal, complemented
3	C₂×C₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₇×#1808	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₇×Q₈	—	2	1	no	—
6	C₂²×C₇×Q₁₆	—	2	8	no	—
7	C₂²×C₇×Q₁₆	—	2	1	no	—
8	C₂²×C₇×Q₁₆	—	2	2	no	—
9	C₂²×C₇×Q₁₆	C₄	1	4	yes	complemented
10	C₂²×C₇×Q₁₆	C₂²	1	2	yes	—
11	C₂³×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂²×C₇×C₁₆	C₂²	1	2	yes	—
13	C₂×C₇×#107	C₂²	1	2	yes	—
14	C₇×#380	—	2	1	no	—
15	C₇×#358	—	2	8	no	—
16	C₇×C₂²⋊C₁₆	—	2	2	no	—
17	C₇×#340	—	2	1	no	—
18	#12732	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
19	C₂²×C₇×Q₈	—	2	8	no	—
20	C₂²×C₇×Q₈	—	2	4	no	—
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂²×C₇×Q₈	—	4	4	no	—
23	C₂²×C₇×Q₈	—	4	1	no	—
24	C₂²×C₇×Q₈	—	4	2	no	—
25	C₂³×C₄×C₇	—	4	1	no	—
26	C₂³×C₄×C₇	D₄	1	1	yes	characteristic
27	C₂×C₇×Q₁₆	—	2	8	no	—
28	C₂×C₇×Q₁₆	—	2	16	no	—
29	C₂×C₇×Q₁₆	—	2	16	no	—
30	C₂×C₇×Q₁₆	—	2	4	no	—
31	C₂×C₇×Q₁₆	—	2	4	no	—
32	C₂×C₇×Q₁₆	—	2	2	no	—
33	C₂×C₇×Q₁₆	C₂×C₄	1	4	yes	—
34	C₂×C₇×Q₁₆	D₄	1	8	yes	—
35	C₂²×C₇×C₈	—	2	4	no	—
36	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
38	C₂²×C₇×C₈	D₄	1	2	yes	—
39	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₇×C₄⋊C₄	—	2	2	no	—
41	C₂×C₇×C₂²⋊C₄	—	4	1	no	—
42	C₂×C₇×C₁₆	—	2	2	no	—
43	C₂×C₇×C₁₆	—	2	4	no	—
44	C₇×#107	—	2	4	no	—
45	C₇×#95	—	4	1	no	—
46	C₂³×Q₁₆	C₂×C₇	1	1	yes	characteristic
47	C₂×#358	C₂×C₇	1	4	yes	complemented
48	C₂×C₂²⋊C₁₆	C₂×C₇	1	1	yes	characteristic
49	#1808	C₂×C₇	1	1	yes	characteristic
50	C₂⁴×C₇	D₈	1	1	yes	characteristic
51	C₂×C₇×Q₈	—	2	16	no	—
52	C₂×C₇×Q₈	—	2	4	no	—
53	C₂×C₇×Q₈	—	2	8	no	—
54	C₂×C₇×Q₈	—	4	8	no	—
55	C₂×C₇×Q₈	—	4	8	no	—
56	C₂×C₇×Q₈	—	4	16	no	—
57	C₂×C₇×Q₈	—	4	4	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂×C₇×Q₈	—	4	4	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	—	2	2	no	—
62	C₂²×C₄×C₇	—	4	8	no	—
63	C₂²×C₄×C₇	—	4	4	no	—
64	C₂²×C₄×C₇	—	4	2	no	—
65	C₂²×C₄×C₇	—	4	2	no	—
66	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₇	SD₁₆	1	2	yes	—
69	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
70	C₇×Q₁₆	—	2	16	no	—
71	C₇×Q₁₆	—	2	4	no	—
72	C₇×Q₁₆	—	2	8	no	—
73	C₇×Q₁₆	C₂²⋊C₄	1	8	yes	complemented
74	C₂×C₇×C₈	—	2	2	no	—
75	C₂×C₇×C₈	—	2	4	no	—
76	C₂×C₇×C₈	—	2	4	no	—
77	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
78	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
80	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
81	C₇×C₄⋊C₄	—	4	2	no	—
82	C₇×C₄⋊C₄	—	4	2	no	—
83	C₇×C₂²⋊C₄	—	8	2	no	—
84	C₇×C₁₆	—	4	2	no	—
85	C₂³×Q₈	—	2	1	no	—
86	C₂²×Q₁₆	—	2	8	no	—
87	C₂²×Q₁₆	—	2	1	no	—
88	C₂²×Q₁₆	—	2	2	no	—
89	C₂²×Q₁₆	C₄×C₇	1	4	yes	complemented
90	C₂²×Q₁₆	C₂²×C₇	1	2	yes	—
91	C₂³×C₈	C₂²×C₇	1	1	yes	characteristic
92	C₂²×C₁₆	C₂²×C₇	1	2	yes	—
93	C₂×#107	C₂²×C₇	1	2	yes	—
94	#380	—	2	1	no	—
95	#358	—	2	8	no	—
96	C₂²⋊C₁₆	—	2	2	no	—
97	#340	—	2	1	no	—
98	C₂³×C₇	—	2	2	no	—
99	C₂³×C₇	—	2	2	no	—
100	C₂³×C₇	#102	1	2	yes	—
101	C₂³×C₇	SD₃₂	1	2	yes	—
102	C₂³×C₇	Q₃₂	1	2	yes	—
103	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
104	C₇×Q₈	—	2	8	no	—
105	C₇×Q₈	—	4	16	no	—
106	C₇×Q₈	—	4	4	no	—
107	C₇×Q₈	—	4	8	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	4	no	—
112	C₂×C₄×C₇	—	4	16	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	—	4	8	no	—
115	C₂×C₄×C₇	—	8	2	no	—
116	C₂×C₄×C₇	—	8	2	no	—
117	C₂×C₄×C₇	—	8	2	no	—
118	C₂×C₄×C₇	#102	1	2	yes	—
119	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	#120	1	2	yes	—
121	C₂×C₄×C₇	C₂×SD₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₇×C₈	—	2	2	no	—
124	C₇×C₈	C₂×C₂²⋊C₄	1	2	yes	—
125	C₇×C₈	#120	1	2	yes	—
126	C₂²×Q₈	—	2	8	no	—
127	C₂²×Q₈	—	2	4	no	—
128	C₂²×Q₈	—	2	2	no	—
129	C₂²×Q₈	—	4	4	no	—
130	C₂²×Q₈	—	4	1	no	—
131	C₂²×Q₈	—	4	2	no	—
132	C₂³×C₄	—	4	1	no	—
133	C₂³×C₄	C₇×D₄	1	1	yes	characteristic
134	C₂×Q₁₆	—	2	8	no	—
135	C₂×Q₁₆	—	2	16	no	—
136	C₂×Q₁₆	—	2	16	no	—
137	C₂×Q₁₆	—	2	4	no	—
138	C₂×Q₁₆	—	2	4	no	—
139	C₂×Q₁₆	—	2	2	no	—
140	C₂×Q₁₆	C₂×C₄×C₇	1	4	yes	—
141	C₂×Q₁₆	C₇×D₄	1	8	yes	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₂×C₄×C₇	1	2	yes	—
144	C₂²×C₈	C₇×D₄	1	1	yes	characteristic
145	C₂²×C₈	C₇×D₄	1	2	yes	—
146	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₂²⋊C₄	—	4	1	no	—
149	C₂×C₁₆	—	2	2	no	—
150	C₂×C₁₆	—	2	4	no	—
151	#107	—	2	4	no	—
152	#95	—	4	1	no	—
153	C₂²×C₇	—	2	2	no	—
154	C₂²×C₇	—	2	4	no	—
155	C₂²×C₇	—	2	4	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	#358	1	4	yes	complemented
158	C₂²×C₇	C₂×#102	1	1	yes	characteristic
159	C₂²×C₇	#447	1	2	yes	—
160	C₂²×C₇	C₂×SD₃₂	1	1	yes	characteristic
161	C₂²×C₇	C₂×Q₃₂	1	1	yes	characteristic
162	C₂²×C₇	#510	1	2	yes	—
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	4	8	no	—
165	C₄×C₇	—	16	2	no	—
166	C₄×C₇	#379	1	1	yes	characteristic
167	C₄×C₇	#418	1	1	yes	characteristic
168	C₄×C₇	#450	1	2	yes	—
169	C₂⁴	C₇×D₈	1	1	yes	characteristic
170	C₂×Q₈	—	2	16	no	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	—	2	8	no	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×Q₈	—	4	8	no	—
175	C₂×Q₈	—	4	16	no	—
176	C₂×Q₈	—	4	4	no	—
177	C₂×Q₈	—	4	2	no	—
178	C₂×Q₈	—	4	4	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	2	2	no	—
181	C₂²×C₄	—	4	8	no	—
182	C₂²×C₄	—	4	4	no	—
183	C₂²×C₄	—	4	2	no	—
184	C₂²×C₄	—	4	2	no	—
185	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
186	C₂²×C₄	C₇×D₈	1	1	yes	characteristic
187	C₂²×C₄	C₇×SD₁₆	1	2	yes	—
188	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
189	Q₁₆	—	2	16	no	—
190	Q₁₆	—	2	4	no	—
191	Q₁₆	—	2	8	no	—
192	Q₁₆	C₇×C₂²⋊C₄	1	8	yes	complemented
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
198	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
199	C₂×C₈	C₂×C₇×D₄	1	1	yes	characteristic
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₂²⋊C₄	—	8	2	no	—
203	C₁₆	—	4	2	no	—
204	C₂×C₇	—	2	4	no	—
205	C₂×C₇	#1786	1	1	yes	characteristic
206	C₂×C₇	C₂×#358	1	1	yes	characteristic
207	C₂×C₇	#2141	1	1	yes	characteristic
208	C₂×C₇	#2185	1	2	yes	—
209	C₂×C₇	#2188	1	2	yes	—
210	C₂³	—	2	2	no	—
211	C₂³	—	2	2	no	—
212	C₂³	C₇×#102	1	2	yes	—
213	C₂³	C₇×SD₃₂	1	2	yes	—
214	C₂³	C₇×Q₃₂	1	2	yes	—
215	C₂³	C₂×C₇×D₈	1	1	yes	characteristic
216	Q₈	—	2	8	no	—
217	Q₈	—	4	16	no	—
218	Q₈	—	4	4	no	—
219	Q₈	—	4	8	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	4	16	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	8	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	2	no	—
230	C₂×C₄	C₇×#102	1	2	yes	—
231	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
232	C₂×C₄	C₇×#120	1	2	yes	—
233	C₂×C₄	C₂×C₇×SD₁₆	1	1	yes	characteristic
234	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
235	C₈	—	2	2	no	—
236	C₈	C₂×C₇×C₂²⋊C₄	1	2	yes	—
237	C₈	C₇×#120	1	2	yes	—
238	C₇	#12732	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	4	no	—
242	C₂²	—	2	4	no	—
243	C₂²	C₇×#358	1	4	yes	complemented
244	C₂²	C₂×C₇×#102	1	1	yes	characteristic
245	C₂²	C₇×#447	1	2	yes	—
246	C₂²	C₂×C₇×SD₃₂	1	1	yes	characteristic
247	C₂²	C₂×C₇×Q₃₂	1	1	yes	characteristic
248	C₂²	C₇×#510	1	2	yes	—
249	C₄	—	2	2	no	—
250	C₄	—	4	8	no	—
251	C₄	—	16	2	no	—
252	C₄	C₇×#379	1	1	yes	characteristic
253	C₄	C₇×#418	1	1	yes	characteristic
254	C₄	C₇×#450	1	2	yes	—
255	C₂	—	2	4	no	—
256	C₂	C₇×#1786	1	1	yes	characteristic
257	C₂	C₂×C₇×#358	1	1	yes	characteristic
258	C₂	C₇×#2141	1	1	yes	characteristic
259	C₂	C₇×#2185	1	2	yes	—
260	C₂	C₇×#2188	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₇
Frattini, cosocle	C₂²×C₈	C₂³×C₇
prime radical	C₂⁴×C₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇ ◁ C₂³×C₇×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₇×#358	C₂	semidirect
9. C₂²×C₇×Q₁₆	C₄	semidirect
18. #12732	C₇	direct (N × H)
47. C₂×#358	C₂×C₇	semidirect
73. C₇×Q₁₆	C₂²⋊C₄	semidirect
89. C₂²×Q₁₆	C₄×C₇	semidirect
157. C₂²×C₇	#358	semidirect
192. Q₁₆	C₇×C₂²⋊C₄	semidirect
238. C₇	#12732	direct (N × H)
243. C₂²	C₇×#358	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₇×Q₁₆	2
C₂×C₇×#358	2
C₂×C₇×C₂²⋊C₁₆	2
C₇×#1808	2
#12732	7

Subgroups — C7×#12732 #13189948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12732 #13189948