Subgroups — C₇×#9708 #13186924

Order 1792.

486 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2862	C₂	1	1	yes	maximal, characteristic
2	C₇×#1842	C₂	1	1	yes	maximal, characteristic
3	C₇×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇×#332	C₂	1	2	yes	maximal
5	C₂×C₇×#332	C₂	1	2	yes	maximal
6	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₇×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₇×C₄⋊C₈	—	2	2	no	—
9	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₇×#97	—	2	1	no	—
12	C₇×#391	—	2	1	no	—
13	C₇×#391	—	2	1	no	—
14	C₇×#391	—	2	2	no	—
15	C₇×#382	C₄	1	4	yes	—
16	C₄³×C₇	C₂²	1	1	yes	characteristic
17	C₇×#336	—	2	1	no	—
18	C₇×#332	—	2	4	no	—
19	C₇×#332	—	2	4	no	—
20	#9708	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	1	no	—
22	C₂²×C₇×Q₈	—	2	1	no	—
23	C₂²×C₇×C₈	—	2	2	no	—
24	C₂²×C₇×C₈	—	2	2	no	—
25	C₇×C₄⋊Q₈	—	2	2	no	—
26	C₇×C₄⋊Q₈	—	2	2	no	—
27	C₇×C₄⋊Q₈	—	2	2	no	—
28	C₇×C₄⋊Q₈	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	2	1	no	—
30	C₂×C₇×C₄⋊C₄	—	2	1	no	—
31	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	D₄	1	2	yes	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	2	no	—
39	C₇×C₄⋊C₈	—	2	2	no	—
40	C₇×C₄⋊C₈	—	2	2	no	—
41	C₇×C₄⋊C₈	—	4	4	no	—
42	C₇×C₄⋊C₈	D₄	1	4	yes	—
43	C₇×#97	—	4	2	no	—
44	C₇×#95	—	2	4	no	—
45	C₇×#95	—	2	4	no	—
46	#2862	C₂×C₇	1	1	yes	characteristic
47	#1842	C₂×C₇	1	1	yes	characteristic
48	#1767	C₂×C₇	1	1	yes	characteristic
49	C₂×#332	C₂×C₇	1	2	yes	—
50	C₂×#332	C₂×C₇	1	2	yes	—
51	C₂×C₇×Q₈	—	4	2	no	—
52	C₂×C₇×Q₈	—	4	1	no	—
53	C₂×C₇×Q₈	—	4	1	no	—
54	C₂×C₇×Q₈	—	4	2	no	—
55	C₂×C₇×Q₈	—	4	1	no	—
56	C₂×C₇×Q₈	—	4	2	no	—
57	C₂×C₇×Q₈	—	4	1	no	—
58	C₂×C₇×Q₈	—	4	2	no	—
59	C₂²×C₄×C₇	—	2	2	no	—
60	C₂²×C₄×C₇	—	2	2	no	—
61	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₇×C₈	—	2	4	no	—
67	C₂×C₇×C₈	—	4	2	no	—
68	C₂×C₇×C₈	—	4	2	no	—
69	C₂×C₇×C₈	—	4	2	no	—
70	C₂×C₇×C₈	—	4	4	no	—
71	C₇×C₄⋊C₄	—	2	4	no	—
72	C₇×C₄⋊C₄	—	2	4	no	—
73	C₇×C₄⋊C₄	—	4	2	no	—
74	C₇×C₄⋊C₄	—	4	1	no	—
75	C₇×C₄⋊C₄	—	4	1	no	—
76	C₄²×C₇	—	2	4	no	—
77	C₄²×C₇	—	2	2	no	—
78	C₄²×C₇	—	2	1	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₄²×C₇	—	2	2	no	—
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	C₂×D₄	1	2	yes	—
83	C₄²×C₇	#41	1	2	yes	—
84	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
85	C₂×C₄⋊Q₈	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₄⋊C₈	—	2	2	no	—
87	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
88	C₂×C₄⋊C₈	C₂²×C₇	1	2	yes	—
89	C₂×#97	—	2	1	no	—
90	#391	—	2	1	no	—
91	#391	—	2	1	no	—
92	#391	—	2	2	no	—
93	#382	C₄×C₇	1	4	yes	—
94	C₄³	C₂²×C₇	1	1	yes	characteristic
95	#336	—	2	1	no	—
96	#332	—	2	4	no	—
97	#332	—	2	4	no	—
98	C₂³×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₇×Q₈	—	8	2	no	—
100	C₇×Q₈	—	8	2	no	—
101	C₇×Q₈	—	8	2	no	—
102	C₇×Q₈	—	8	1	no	—
103	C₇×Q₈	—	8	1	no	—
104	C₂×C₄×C₇	—	2	2	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	4	2	no	—
111	C₂×C₄×C₇	—	4	2	no	—
112	C₂×C₄×C₇	—	4	2	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	#101	1	4	yes	—
116	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
117	C₂×C₄×C₇	#121	1	2	yes	—
118	C₂×C₄×C₇	#124	1	1	yes	characteristic
119	C₂×C₄×C₇	#127	1	1	yes	characteristic
120	C₂×C₄×C₇	#135	1	2	yes	—
121	C₂×C₄×C₇	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄×C₇	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₇	#137	1	1	yes	characteristic
124	C₂×C₄×C₇	#137	1	2	yes	—
125	C₇×C₈	—	4	4	no	—
126	C₇×C₈	—	8	2	no	—
127	C₂²×Q₈	—	2	1	no	—
128	C₂²×Q₈	—	2	1	no	—
129	C₂²×C₈	—	2	2	no	—
130	C₂²×C₈	—	2	2	no	—
131	C₄⋊Q₈	—	2	2	no	—
132	C₄⋊Q₈	—	2	2	no	—
133	C₄⋊Q₈	—	2	2	no	—
134	C₄⋊Q₈	—	2	2	no	—
135	C₂×C₄⋊C₄	—	2	1	no	—
136	C₂×C₄⋊C₄	—	2	1	no	—
137	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
138	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
139	C₂×C₄²	C₂×C₄×C₇	1	2	yes	—
140	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
141	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
142	C₂×C₄²	C₇×D₄	1	2	yes	—
143	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
144	C₄⋊C₈	—	2	2	no	—
145	C₄⋊C₈	—	2	2	no	—
146	C₄⋊C₈	—	2	2	no	—
147	C₄⋊C₈	—	4	4	no	—
148	C₄⋊C₈	C₇×D₄	1	4	yes	—
149	#97	—	4	2	no	—
150	#95	—	2	4	no	—
151	#95	—	2	4	no	—
152	C₂²×C₇	#390	1	1	yes	characteristic
153	C₂²×C₇	C₂×#101	1	2	yes	—
154	C₂²×C₇	#416	1	1	yes	characteristic
155	C₂²×C₇	#417	1	2	yes	—
156	C₂²×C₇	#417	1	1	yes	characteristic
157	C₄×C₇	—	2	4	no	—
158	C₄×C₇	—	2	2	no	—
159	C₄×C₇	—	4	2	no	—
160	C₄×C₇	—	8	2	no	—
161	C₄×C₇	—	8	2	no	—
162	C₄×C₇	#463	1	2	yes	—
163	C₄×C₇	#464	1	2	yes	—
164	C₄×C₇	#489	1	2	yes	—
165	C₄×C₇	#496	1	1	yes	characteristic
166	C₄×C₇	#497	1	1	yes	characteristic
167	C₂×Q₈	—	4	2	no	—
168	C₂×Q₈	—	4	1	no	—
169	C₂×Q₈	—	4	1	no	—
170	C₂×Q₈	—	4	2	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
179	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	4	4	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	2	4	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄⋊C₄	—	4	1	no	—
192	C₄²	—	2	4	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	—	2	2	no	—
196	C₄²	—	2	2	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₂×C₇×D₄	1	2	yes	—
199	C₄²	C₇×#41	1	2	yes	—
200	C₂×C₇	#1552	1	2	yes	—
201	C₂×C₇	#1561	1	2	yes	—
202	C₂×C₇	#1959	1	1	yes	characteristic
203	C₂×C₇	#1966	1	1	yes	characteristic
204	C₂×C₇	#1983	1	1	yes	characteristic
205	C₂³	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
206	Q₈	—	8	2	no	—
207	Q₈	—	8	1	no	—
208	Q₈	—	8	1	no	—
209	Q₈	—	8	2	no	—
210	Q₈	—	8	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	C₇×#101	1	4	yes	—
223	C₂×C₄	C₄×C₇×D₄	1	2	yes	—
224	C₂×C₄	C₇×#121	1	2	yes	—
225	C₂×C₄	C₇×#124	1	1	yes	characteristic
226	C₂×C₄	C₇×#127	1	1	yes	characteristic
227	C₂×C₄	C₇×#135	1	2	yes	—
228	C₂×C₄	C₇×AGL(1,ℤ/8)	1	2	yes	—
229	C₂×C₄	C₇×AGL(1,ℤ/8)	1	1	yes	characteristic
230	C₂×C₄	C₇×#137	1	2	yes	—
231	C₂×C₄	C₇×#137	1	1	yes	characteristic
232	C₈	—	4	4	no	—
233	C₈	—	8	2	no	—
234	C₇	#9708	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²	C₇×#390	1	1	yes	characteristic
236	C₂²	C₂×C₇×#101	1	2	yes	—
237	C₂²	C₇×#416	1	1	yes	characteristic
238	C₂²	C₇×#417	1	1	yes	characteristic
239	C₂²	C₇×#417	1	2	yes	—
240	C₄	—	2	2	no	—
241	C₄	—	2	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₇×#463	1	2	yes	—
246	C₄	C₇×#464	1	2	yes	—
247	C₄	C₇×#489	1	2	yes	—
248	C₄	C₇×#496	1	1	yes	characteristic
249	C₄	C₇×#497	1	1	yes	characteristic
250	C₂	C₇×#1552	1	2	yes	—
251	C₂	C₇×#1561	1	2	yes	—
252	C₂	C₇×#1959	1	1	yes	characteristic
253	C₂	C₇×#1966	1	1	yes	characteristic
254	C₂	C₇×#1983	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₇	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₇
Frattini, cosocle	C₂×C₄²	C₂³×C₇
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. #9708	C₇	direct (N × H)
234. C₇	#9708	direct (N × H)

Maximal subgroups

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

subgroup	index
C₇×#2862	2
C₇×#1842	2
C₇×#1767	2
C₂×C₇×#332	2
#9708	7

Subgroups — C7×#9708 #13186924

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#9708 #13186924