Subgroups — C₇×#12849 #13190065

Order 1792.

462 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₇×#349	C₂	1	2	yes	maximal
3	C₇×#1376	C₂	1	2	yes	maximal, complemented
4	C₇×#1375	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₂×C₇×M₃₂	C₂²	1	4	yes	—
9	C₄×C₇×M₁₆	—	2	4	no	—
10	C₂×C₇×#97	C₄	1	1	yes	characteristic
11	C₂×C₇×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₇×#349	—	2	4	no	—
15	#12849	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
16	C₂³×C₄×C₇	—	2	1	no	—
17	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₇×M₁₆	—	2	2	no	—
19	C₂×C₇×M₁₆	—	2	2	no	—
20	C₂×C₇×M₁₆	—	2	2	no	—
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	D₄	1	4	yes	—
24	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₇	—	2	2	no	—
29	C₂×C₄²×C₇	—	2	1	no	—
30	C₂×C₄²×C₇	C₈	1	4	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₇×M₃₂	—	2	4	no	—
34	C₇×M₃₂	—	2	4	no	—
35	C₂×C₇×C₁₆	—	2	4	no	—
36	C₇×#97	—	2	2	no	—
37	C₇×#97	—	2	2	no	—
38	C₄×C₇×C₈	—	2	2	no	—
39	C₄×C₇×C₈	—	2	2	no	—
40	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
41	C₂×#349	C₂×C₇	1	2	yes	—
42	#1376	C₂×C₇	1	2	yes	complemented
43	#1375	C₂×C₇	1	2	yes	complemented
44	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
45	C₂²×C₄×C₇	—	2	1	no	—
46	C₂²×C₄×C₇	—	2	1	no	—
47	C₂²×C₄×C₇	—	2	1	no	—
48	C₂²×C₄×C₇	—	2	1	no	—
49	C₂²×C₄×C₇	—	2	1	no	—
50	C₂²×C₄×C₇	—	2	4	no	—
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	4	4	no	—
53	C₂²×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
55	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
56	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
57	C₇×M₁₆	—	2	8	no	—
58	C₇×M₁₆	—	2	8	no	—
59	C₂×C₇×C₈	—	2	1	no	—
60	C₂×C₇×C₈	—	2	1	no	—
61	C₂×C₇×C₈	—	2	1	no	—
62	C₂×C₇×C₈	—	2	1	no	—
63	C₂×C₇×C₈	—	2	1	no	—
64	C₂×C₇×C₈	—	2	1	no	—
65	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
66	C₂×C₇×C₈	M₁₆	1	2	yes	—
67	C₂×C₇×C₈	M₁₆	1	2	yes	—
68	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
69	C₂×C₇×C₈	#41	1	2	yes	—
70	C₂×C₇×C₈	#41	1	2	yes	—
71	C₄²×C₇	—	2	4	no	—
72	C₄²×C₇	—	2	1	no	—
73	C₄²×C₇	—	2	1	no	—
74	C₄²×C₇	C₂×C₈	1	2	yes	—
75	C₄²×C₇	C₂×C₈	1	2	yes	—
76	C₇×C₁₆	—	4	4	no	—
77	C₂²×M₁₆	C₄×C₇	1	1	yes	characteristic
78	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
79	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
80	C₂×M₃₂	C₂²×C₇	1	4	yes	—
81	C₄×M₁₆	—	2	4	no	—
82	C₂×#97	C₄×C₇	1	1	yes	characteristic
83	C₂×#97	C₂²×C₇	1	1	yes	characteristic
84	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
85	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
86	#349	—	2	4	no	—
87	C₂³×C₇	—	2	1	no	—
88	C₂³×C₇	—	2	1	no	—
89	C₂³×C₇	—	4	2	no	—
90	C₂³×C₇	C₂²⋊C₈	1	2	yes	—
91	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₇	—	2	4	no	—
93	C₂×C₄×C₇	—	2	2	no	—
94	C₂×C₄×C₇	—	2	1	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	2	no	—
97	C₂×C₄×C₇	—	2	2	no	—
98	C₂×C₄×C₇	—	2	2	no	—
99	C₂×C₄×C₇	—	2	2	no	—
100	C₂×C₄×C₇	—	2	2	no	—
101	C₂×C₄×C₇	—	4	8	no	—
102	C₂×C₄×C₇	—	4	2	no	—
103	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇	#117	1	1	yes	characteristic
106	C₂×C₄×C₇	#117	1	1	yes	characteristic
107	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
110	C₇×C₈	—	2	2	no	—
111	C₇×C₈	—	2	2	no	—
112	C₇×C₈	—	2	2	no	—
113	C₇×C₈	#123	1	4	yes	—
114	C₂³×C₄	—	2	1	no	—
115	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
116	C₂×M₁₆	—	2	2	no	—
117	C₂×M₁₆	—	2	2	no	—
118	C₂×M₁₆	—	2	2	no	—
119	C₂×M₁₆	—	2	2	no	—
120	C₂×M₁₆	—	2	2	no	—
121	C₂×M₁₆	C₇×D₄	1	4	yes	—
122	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
123	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
124	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
125	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
126	C₂×C₄²	—	2	2	no	—
127	C₂×C₄²	—	2	1	no	—
128	C₂×C₄²	C₇×C₈	1	4	yes	—
129	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
130	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
131	M₃₂	—	2	4	no	—
132	M₃₂	—	2	4	no	—
133	C₂×C₁₆	—	2	4	no	—
134	#97	—	2	2	no	—
135	#97	—	2	2	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	2	2	no	—
138	C₂²×C₇	—	2	2	no	—
139	C₂²×C₇	—	2	2	no	—
140	C₂²×C₇	—	4	2	no	—
141	C₂²×C₇	—	4	2	no	—
142	C₂²×C₇	—	4	2	no	—
143	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₇	#432	1	1	yes	characteristic
145	C₂²×C₇	#432	1	1	yes	characteristic
146	C₄×C₇	—	2	4	no	—
147	C₄×C₇	—	2	2	no	—
148	C₄×C₇	—	4	2	no	—
149	C₄×C₇	#381	1	1	yes	characteristic
150	C₄×C₇	#407	1	1	yes	characteristic
151	C₄×C₇	#431	1	1	yes	characteristic
152	C₄×C₇	#431	1	1	yes	characteristic
153	C₂⁴	C₇×M₁₆	1	1	yes	characteristic
154	C₂²×C₄	—	2	1	no	—
155	C₂²×C₄	—	2	1	no	—
156	C₂²×C₄	—	2	1	no	—
157	C₂²×C₄	—	2	1	no	—
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	4	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	4	4	no	—
162	C₂²×C₄	C₇×C₂²⋊C₄	1	2	yes	—
163	C₂²×C₄	C₂×C₇×C₈	1	2	yes	—
164	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
166	M₁₆	—	2	8	no	—
167	M₁₆	—	2	8	no	—
168	C₂×C₈	—	2	1	no	—
169	C₂×C₈	—	2	1	no	—
170	C₂×C₈	—	2	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	C₇×C₂²⋊C₄	1	2	yes	—
175	C₂×C₈	C₇×M₁₆	1	2	yes	—
176	C₂×C₈	C₇×M₁₆	1	2	yes	—
177	C₂×C₈	C₂×C₇×D₄	1	2	yes	—
178	C₂×C₈	C₇×#41	1	2	yes	—
179	C₂×C₈	C₇×#41	1	2	yes	—
180	C₄²	—	2	4	no	—
181	C₄²	—	2	1	no	—
182	C₄²	—	2	1	no	—
183	C₄²	C₂×C₇×C₈	1	2	yes	—
184	C₄²	C₂×C₇×C₈	1	2	yes	—
185	C₁₆	—	4	4	no	—
186	C₂×C₇	—	2	1	no	—
187	C₂×C₇	—	2	1	no	—
188	C₂×C₇	—	4	2	no	—
189	C₂×C₇	#1817	1	1	yes	characteristic
190	C₂×C₇	#2165	1	2	yes	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	C₇×C₂²⋊C₈	1	2	yes	—
195	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	8	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	C₇×C₂²⋊C₈	1	2	yes	—
208	C₂×C₄	C₂×C₇×C₂²⋊C₄	1	1	yes	characteristic
209	C₂×C₄	C₇×#117	1	1	yes	characteristic
210	C₂×C₄	C₇×#117	1	1	yes	characteristic
211	C₂×C₄	C₂²×C₇×C₈	1	1	yes	characteristic
212	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
213	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	—	2	2	no	—
217	C₈	C₇×#123	1	4	yes	—
218	C₇	#12849	1	1	yes	characteristic, direct product factor, Sylow
219	C₂²	—	2	2	no	—
220	C₂²	—	2	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	—	4	2	no	—
223	C₂²	—	4	2	no	—
224	C₂²	C₂×C₇×C₂²⋊C₈	1	1	yes	characteristic
225	C₂²	C₇×#432	1	1	yes	characteristic
226	C₂²	C₇×#432	1	1	yes	characteristic
227	C₄	—	2	4	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	C₇×#381	1	1	yes	characteristic
231	C₄	C₇×#407	1	1	yes	characteristic
232	C₄	C₇×#431	1	1	yes	characteristic
233	C₄	C₇×#431	1	1	yes	characteristic
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	—	4	2	no	—
237	C₂	C₇×#1817	1	1	yes	characteristic
238	C₂	C₇×#2165	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂×C₄	C₂²×C₇×C₈
Frattini, cosocle	C₂²×C₈	C₂³×C₇
socle	C₂²×C₇	#432
prime radical	C₂⁴×C₇	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₇ ◁ C₂×C₇×#97 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₇×#1376	C₂	semidirect
4. C₇×#1375	C₂	semidirect
15. #12849	C₇	direct (N × H)
42. #1376	C₂×C₇	semidirect
43. #1375	C₂×C₇	semidirect
218. C₇	#12849	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₇×M₁₆	2
C₂×C₇×#349	2
C₇×#1376	2
C₇×#1375	2
#12849	7

Subgroups — C7×#12849 #13190065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#12849 #13190065