Subgroups — C₇×#17897 #13195113

Order 1792.

614 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×#2983	C₂	1	2	yes	maximal
2	C₂×C₄×C₇×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₇×#1593	C₂	1	4	yes	maximal, complemented
4	C₇×#1591	C₂	1	4	yes	maximal, complemented
5	C₇×#1564	C₂	1	1	yes	maximal, characteristic
6	C₇×#1479	C₂	1	2	yes	maximal
7	C₇×#1450	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₇×Q₁₆	C₄	1	1	yes	characteristic
9	C₂×C₄×C₇×Q₈	C₂²	1	2	yes	—
10	C₇×#446	—	2	4	no	—
11	C₇×#446	C₂²	1	8	yes	—
12	C₄×C₇×Q₁₆	C₄	1	2	yes	—
13	C₄×C₇×Q₁₆	C₄	1	2	yes	—
14	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
15	C₄×C₇×Q₁₆	C₂²	1	2	yes	—
16	C₇×#432	—	2	4	no	—
17	C₇×#431	C₂²	1	2	yes	—
18	C₇×#431	C₂²	1	1	yes	characteristic
19	C₇×#431	C₂²	1	1	yes	characteristic
20	C₂×C₇×#107	C₄	1	1	yes	characteristic
21	C₂×C₇×#103	C₄	1	2	yes	—
22	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
23	C₇×#335	C₂²	1	2	yes	—
24	C₇×#334	C₂²	1	2	yes	—
25	C₇×Q₈⋊C₈	C₂²	1	8	yes	—
26	C₇×#322	C₂²	1	2	yes	—
27	C₇×C₈²	C₂²	1	2	yes	—
28	#17897	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
29	C₂²×C₇×Q₈	C₂×C₄	1	2	yes	—
30	C₂×C₇×Q₁₆	—	2	4	no	—
31	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
32	C₂×C₇×Q₁₆	C₂×C₄	1	2	yes	—
33	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
35	C₄×C₇×Q₈	—	2	2	no	—
36	C₄×C₇×Q₈	—	2	2	no	—
37	C₄×C₇×Q₈	C₂×C₄	1	4	yes	—
38	C₄×C₇×Q₈	C₂³	1	4	yes	—
39	C₂×C₇×C₄⋊C₄	—	2	2	no	—
40	C₂×C₇×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₇	—	2	2	no	—
42	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
43	C₇×#107	C₂×C₄	1	2	yes	—
44	C₇×#107	C₂×C₄	1	2	yes	—
45	C₇×C₄⋊C₈	—	2	8	no	—
46	C₇×C₄⋊C₈	—	2	1	no	—
47	C₇×C₄⋊C₈	D₄	1	2	yes	—
48	C₇×C₄⋊C₈	C₂³	1	4	yes	—
49	C₇×#103	C₂×C₄	1	4	yes	—
50	C₇×#103	C₂×C₄	1	4	yes	—
51	C₇×C₂²⋊C₈	—	2	2	no	—
52	C₇×C₂²⋊C₈	—	2	1	no	—
53	C₇×C₂²⋊C₈	D₄	1	2	yes	—
54	C₇×#97	—	2	8	no	—
55	C₄×C₇×C₈	—	2	2	no	—
56	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
57	C₄×C₇×C₈	C₂×C₄	1	1	yes	characteristic
58	C₄×C₇×C₈	C₂³	1	2	yes	—
59	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
60	C₄×C₇×C₈	C₂³	1	1	yes	characteristic
61	C₄×C₇×C₈	C₂³	1	2	yes	—
62	#2983	C₂×C₇	1	2	yes	—
63	C₂×C₄×Q₁₆	C₂×C₇	1	1	yes	characteristic
64	#1593	C₂×C₇	1	4	yes	complemented
65	#1591	C₂×C₇	1	4	yes	complemented
66	#1564	C₂×C₇	1	1	yes	characteristic
67	#1479	C₂×C₇	1	2	yes	—
68	#1450	C₂×C₇	1	1	yes	characteristic
69	C₂×C₇×Q₈	—	2	2	no	—
70	C₂×C₇×Q₈	—	2	4	no	—
71	C₂×C₇×Q₈	—	2	4	no	—
72	C₂×C₇×Q₈	C₂²×C₄	1	4	yes	—
73	C₂²×C₄×C₇	—	2	2	no	—
74	C₂²×C₄×C₇	—	2	2	no	—
75	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇	#41	1	1	yes	characteristic
78	C₇×Q₁₆	—	2	4	no	—
79	C₇×Q₁₆	—	2	4	no	—
80	C₂×C₇×C₈	—	2	2	no	—
81	C₂×C₇×C₈	—	2	4	no	—
82	C₂×C₇×C₈	—	2	2	no	—
83	C₂×C₇×C₈	M₁₆	1	4	yes	—
84	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
89	C₇×C₄⋊C₄	—	2	4	no	—
90	C₇×C₄⋊C₄	—	2	2	no	—
91	C₇×C₄⋊C₄	—	2	4	no	—
92	C₇×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₄²×C₇	—	2	4	no	—
94	C₄²×C₇	—	2	4	no	—
95	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₇	#41	1	1	yes	characteristic
98	C₄²×C₇	C₂⁴	1	1	yes	characteristic
99	C₂²×Q₁₆	C₄×C₇	1	1	yes	characteristic
100	C₂×C₄×Q₈	C₂²×C₇	1	2	yes	—
101	#446	—	2	4	no	—
102	#446	C₂²×C₇	1	8	yes	—
103	C₄×Q₁₆	C₄×C₇	1	2	yes	—
104	C₄×Q₁₆	C₄×C₇	1	2	yes	—
105	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
106	C₄×Q₁₆	C₂²×C₇	1	2	yes	—
107	#432	—	2	4	no	—
108	#431	C₂²×C₇	1	2	yes	—
109	#431	C₂²×C₇	1	1	yes	characteristic
110	#431	C₂²×C₇	1	1	yes	characteristic
111	C₂×#107	C₄×C₇	1	1	yes	characteristic
112	C₂×#103	C₄×C₇	1	2	yes	—
113	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
114	#335	C₂²×C₇	1	2	yes	—
115	#334	C₂²×C₇	1	2	yes	—
116	Q₈⋊C₈	C₂²×C₇	1	8	yes	—
117	#322	C₂²×C₇	1	2	yes	—
118	C₈²	C₂²×C₇	1	2	yes	—
119	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
120	C₇×Q₈	—	2	4	no	—
121	C₇×Q₈	—	2	4	no	—
122	C₇×Q₈	—	4	4	no	—
123	C₂×C₄×C₇	—	2	2	no	—
124	C₂×C₄×C₇	—	2	4	no	—
125	C₂×C₄×C₇	—	2	4	no	—
126	C₂×C₄×C₇	—	4	1	no	—
127	C₂×C₄×C₇	—	4	4	no	—
128	C₂×C₄×C₇	—	8	2	no	—
129	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
133	C₂×C₄×C₇	#131	1	2	yes	—
134	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
137	C₇×C₈	—	2	1	no	—
138	C₇×C₈	—	2	1	no	—
139	C₇×C₈	—	2	4	no	—
140	C₇×C₈	—	4	2	no	—
141	C₇×C₈	—	4	4	no	—
142	C₇×C₈	C₂×M₁₆	1	2	yes	—
143	C₇×C₈	C₂×M₁₆	1	2	yes	—
144	C₂²×Q₈	C₂×C₄×C₇	1	2	yes	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
147	C₂×Q₁₆	C₂×C₄×C₇	1	2	yes	—
148	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
149	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
150	C₄×Q₈	—	2	2	no	—
151	C₄×Q₈	—	2	2	no	—
152	C₄×Q₈	C₂×C₄×C₇	1	4	yes	—
153	C₄×Q₈	C₂³×C₇	1	4	yes	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	C₂×C₄×C₇	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
158	#107	C₂×C₄×C₇	1	2	yes	—
159	#107	C₂×C₄×C₇	1	2	yes	—
160	C₄⋊C₈	—	2	8	no	—
161	C₄⋊C₈	—	2	1	no	—
162	C₄⋊C₈	C₇×D₄	1	2	yes	—
163	C₄⋊C₈	C₂³×C₇	1	4	yes	—
164	#103	C₂×C₄×C₇	1	4	yes	—
165	#103	C₂×C₄×C₇	1	4	yes	—
166	C₂²⋊C₈	—	2	2	no	—
167	C₂²⋊C₈	—	2	1	no	—
168	C₂²⋊C₈	C₇×D₄	1	2	yes	—
169	#97	—	2	8	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
172	C₄×C₈	C₂×C₄×C₇	1	1	yes	characteristic
173	C₄×C₈	C₂³×C₇	1	2	yes	—
174	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
175	C₄×C₈	C₂³×C₇	1	1	yes	characteristic
176	C₄×C₈	C₂³×C₇	1	2	yes	—
177	C₂²×C₇	—	2	1	no	—
178	C₂²×C₇	—	2	1	no	—
179	C₂²×C₇	#436	1	2	yes	—
180	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
181	C₄×C₇	—	2	1	no	—
182	C₄×C₇	—	2	1	no	—
183	C₄×C₇	—	2	1	no	—
184	C₄×C₇	—	4	4	no	—
185	C₄×C₇	—	4	4	no	—
186	C₄×C₇	#436	1	2	yes	—
187	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
188	C₄×C₇	C₂×#131	1	1	yes	characteristic
189	C₄×C₇	#578	1	2	yes	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	C₂²×C₄×C₇	1	4	yes	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
197	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
198	C₂²×C₄	C₇×#41	1	1	yes	characteristic
199	Q₁₆	—	2	4	no	—
200	Q₁₆	—	2	4	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₇×M₁₆	1	4	yes	—
205	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
206	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
207	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
208	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
209	C₂×C₈	C₂×C₇×D₄	1	4	yes	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	C₂²×C₄×C₇	1	4	yes	—
214	C₄²	—	2	4	no	—
215	C₄²	—	2	4	no	—
216	C₄²	C₂²×C₄×C₇	1	1	yes	characteristic
217	C₄²	C₂×C₇×D₄	1	1	yes	characteristic
218	C₄²	C₇×#41	1	1	yes	characteristic
219	C₄²	C₂⁴×C₇	1	1	yes	characteristic
220	C₂×C₇	—	2	2	no	—
221	C₂×C₇	C₂×#436	1	1	yes	characteristic
222	C₂×C₇	#2950	1	1	yes	characteristic
223	C₂×C₇	#2957	1	1	yes	characteristic
224	C₂³	C₄×C₇×D₄	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	2	4	no	—
227	Q₈	—	4	4	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
235	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
236	C₂×C₄	C₄×C₇×D₄	1	1	yes	characteristic
237	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
238	C₂×C₄	C₇×#131	1	2	yes	—
239	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
240	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
241	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
242	C₈	—	2	1	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	4	no	—
245	C₈	—	4	2	no	—
246	C₈	—	4	4	no	—
247	C₈	C₂×C₇×M₁₆	1	2	yes	—
248	C₈	C₂×C₇×M₁₆	1	2	yes	—
249	C₇	#17897	1	1	yes	characteristic, direct product factor, Sylow
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	C₇×#436	1	2	yes	—
253	C₂²	C₂×C₄×C₇×D₄	1	1	yes	characteristic
254	C₄	—	2	1	no	—
255	C₄	—	2	1	no	—
256	C₄	—	2	1	no	—
257	C₄	—	4	4	no	—
258	C₄	—	4	4	no	—
259	C₄	C₇×#436	1	2	yes	—
260	C₄	C₂²×C₇×M₁₆	1	1	yes	characteristic
261	C₄	C₂×C₇×#131	1	1	yes	characteristic
262	C₄	C₇×#578	1	2	yes	—
263	C₂	—	2	2	no	—
264	C₂	C₂×C₇×#436	1	1	yes	characteristic
265	C₂	C₇×#2950	1	1	yes	characteristic
266	C₂	C₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₄²	C₂⁴×C₇
socle	C₂²×C₇	C₂×C₄×D₄
prime radical	C₂³×C₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₇×#431 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1593	C₂	semidirect
4. C₇×#1591	C₂	semidirect
28. #17897	C₇	direct (N × H)
64. #1593	C₂×C₇	semidirect
65. #1591	C₂×C₇	semidirect
249. C₇	#17897	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×#2983	2
C₂×C₄×C₇×Q₁₆	2
C₇×#1593	2
C₇×#1591	2
C₇×#1564	2
C₇×#1479	2
C₇×#1450	2
#17897	7

Subgroups — C7×#17897 #13195113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#17897 #13195113