Subgroups — C₂×C₇×#2165 #13210640

Order 1792.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₇×#2165	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₇×#364	C₂	1	2	yes	maximal, complemented
4	C₂×C₇×#349	C₂	1	2	yes	maximal
5	C₂×C₇×#347	C₂	1	2	yes	maximal, complemented
6	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
9	C₂×C₇×M₃₂	C₂²	1	4	yes	—
10	C₄×C₇×M₁₆	C₄	1	4	yes	—
11	C₄×C₇×M₁₆	C₂²	1	4	yes	—
12	C₂×C₇×#97	C₄	1	1	yes	characteristic
13	C₂×C₇×#97	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
15	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
16	C₇×#364	C₂²	1	8	yes	complemented
17	C₇×#349	C₂²	1	8	yes	—
18	C₇×#347	C₂²	1	8	yes	complemented
19	C₂×#2165	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₇	—	2	1	no	—
21	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	—	2	2	no	—
26	C₂×C₇×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
28	C₂×C₇×M₁₆	C₂³	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₇	—	2	1	no	—
34	C₂×C₄²×C₇	—	2	2	no	—
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
38	C₇×M₃₂	—	2	4	no	—
39	C₇×M₃₂	C₂³	1	8	yes	—
40	C₂×C₇×C₁₆	—	2	4	no	—
41	C₇×#97	C₂×C₄	1	4	yes	—
42	C₇×#97	C₂×C₄	1	2	yes	—
43	C₇×#97	C₂³	1	2	yes	—
44	C₄×C₇×C₈	C₂×C₄	1	4	yes	—
45	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₇×C₈	C₂³	1	2	yes	—
47	C₂×C₄×M₁₆	C₂×C₇	1	1	yes	characteristic
48	#2165	C₂×C₇	1	8	yes	direct product factor
49	C₂×#364	C₂×C₇	1	2	yes	complemented
50	C₂×#349	C₂×C₇	1	2	yes	—
51	C₂×#347	C₂×C₇	1	2	yes	complemented
52	C₂⁴×C₇	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	1	no	—
56	C₂²×C₄×C₇	—	2	1	no	—
57	C₂²×C₄×C₇	—	2	1	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	—	2	1	no	—
60	C₂²×C₄×C₇	—	4	4	no	—
61	C₂²×C₄×C₇	M₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₇	M₁₆	1	2	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
65	C₇×M₁₆	—	2	4	no	—
66	C₇×M₁₆	—	2	8	no	—
67	C₇×M₁₆	—	2	4	no	—
68	C₂×C₇×C₈	—	2	1	no	—
69	C₂×C₇×C₈	—	2	1	no	—
70	C₂×C₇×C₈	—	2	1	no	—
71	C₂×C₇×C₈	—	2	1	no	—
72	C₂×C₇×C₈	M₁₆	1	2	yes	—
73	C₂×C₇×C₈	M₁₆	1	2	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
78	C₂×C₇×C₈	#41	1	2	yes	—
79	C₂×C₇×C₈	#41	1	2	yes	—
80	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	—	2	4	no	—
83	C₄²×C₇	—	2	1	no	—
84	C₄²×C₇	C₂²×C₄	1	2	yes	—
85	C₄²×C₇	C₂²×C₄	1	2	yes	—
86	C₇×C₁₆	—	2	8	no	—
87	C₂²×M₁₆	C₄×C₇	1	1	yes	characteristic
88	C₂²×M₁₆	C₂²×C₇	1	1	yes	characteristic
89	C₂²×C₄²	C₄×C₇	1	1	yes	characteristic
90	C₂×M₃₂	C₂²×C₇	1	4	yes	—
91	C₄×M₁₆	C₄×C₇	1	4	yes	—
92	C₄×M₁₆	C₂²×C₇	1	4	yes	—
93	C₂×#97	C₄×C₇	1	1	yes	characteristic
94	C₂×#97	C₂²×C₇	1	1	yes	characteristic
95	C₂×C₄×C₈	C₄×C₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂²×C₇	1	1	yes	characteristic
97	#364	C₂²×C₇	1	8	yes	complemented
98	#349	C₂²×C₇	1	8	yes	—
99	#347	C₂²×C₇	1	8	yes	complemented
100	C₂³×C₇	—	2	1	no	—
101	C₂³×C₇	—	2	1	no	—
102	C₂³×C₇	—	4	2	no	—
103	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
104	C₂³×C₇	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₇	—	2	4	no	—
106	C₂×C₄×C₇	—	2	2	no	—
107	C₂×C₄×C₇	—	2	1	no	—
108	C₂×C₄×C₇	—	2	1	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	2	2	no	—
113	C₂×C₄×C₇	—	2	2	no	—
114	C₂×C₄×C₇	—	4	8	no	—
115	C₂×C₄×C₇	—	4	2	no	—
116	C₂×C₄×C₇	#117	1	1	yes	characteristic
117	C₂×C₄×C₇	#117	1	1	yes	characteristic
118	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
123	C₇×C₈	—	2	2	no	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	#117	1	2	yes	—
126	C₇×C₈	C₂×M₁₆	1	4	yes	—
127	C₇×C₈	C₂×#41	1	2	yes	—
128	C₂³×C₄	—	2	1	no	—
129	C₂³×C₄	C₂×C₄×C₇	1	1	yes	characteristic
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₂×C₄×C₇	1	4	yes	—
135	C₂×M₁₆	C₂×C₄×C₇	1	2	yes	—
136	C₂×M₁₆	C₂³×C₇	1	2	yes	—
137	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
139	C₂²×C₈	C₂×C₄×C₇	1	1	yes	characteristic
140	C₂²×C₈	C₂³×C₇	1	1	yes	characteristic
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	—	2	2	no	—
143	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
144	C₂×C₄²	C₂×C₄×C₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×C₄×C₇	1	4	yes	—
146	M₃₂	—	2	4	no	—
147	M₃₂	C₂³×C₇	1	8	yes	—
148	C₂×C₁₆	—	2	4	no	—
149	#97	C₂×C₄×C₇	1	4	yes	—
150	#97	C₂×C₄×C₇	1	2	yes	—
151	#97	C₂³×C₇	1	2	yes	—
152	C₄×C₈	C₂×C₄×C₇	1	4	yes	—
153	C₄×C₈	C₂×C₄×C₇	1	2	yes	—
154	C₄×C₈	C₂³×C₇	1	2	yes	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	4	2	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	#432	1	1	yes	characteristic
161	C₂²×C₇	#432	1	1	yes	characteristic
162	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
163	C₄×C₇	—	2	4	no	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	4	2	no	—
166	C₄×C₇	#432	1	1	yes	characteristic
167	C₄×C₇	#432	1	1	yes	characteristic
168	C₄×C₇	C₂×#117	1	1	yes	characteristic
169	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
170	C₂⁴	C₇×M₁₆	1	1	yes	characteristic
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	C₇×M₁₆	1	1	yes	characteristic
180	C₂²×C₄	C₇×M₁₆	1	2	yes	—
181	C₂²×C₄	C₂²×C₄×C₇	1	2	yes	—
182	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
183	M₁₆	—	2	4	no	—
184	M₁₆	—	2	8	no	—
185	M₁₆	—	2	4	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	C₇×M₁₆	1	2	yes	—
191	C₂×C₈	C₇×M₁₆	1	2	yes	—
192	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
194	C₂×C₈	C₂²×C₄×C₇	1	1	yes	characteristic
195	C₂×C₈	C₂²×C₄×C₇	1	2	yes	—
196	C₂×C₈	C₇×#41	1	2	yes	—
197	C₂×C₈	C₇×#41	1	2	yes	—
198	C₂×C₈	C₂⁴×C₇	1	1	yes	characteristic
199	C₄²	—	2	1	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	1	no	—
202	C₄²	C₂²×C₄×C₇	1	2	yes	—
203	C₄²	C₂²×C₄×C₇	1	2	yes	—
204	C₁₆	—	2	8	no	—
205	C₂×C₇	—	2	1	no	—
206	C₂×C₇	—	2	1	no	—
207	C₂×C₇	—	4	2	no	—
208	C₂×C₇	#2165	1	2	yes	direct product factor
209	C₂×C₇	C₂×#432	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	—	4	2	no	—
213	C₂³	C₂×C₇×M₁₆	1	1	yes	characteristic
214	C₂³	C₂×C₇×M₁₆	1	2	yes	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₇×#117	1	1	yes	characteristic
227	C₂×C₄	C₇×#117	1	1	yes	characteristic
228	C₂×C₄	C₂×C₇×M₁₆	1	2	yes	—
229	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
230	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
231	C₂×C₄	C₂×C₇×M₁₆	1	1	yes	characteristic
232	C₂×C₄	C₂³×C₄×C₇	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	C₇×#117	1	2	yes	—
236	C₈	C₂×C₇×M₁₆	1	4	yes	—
237	C₈	C₂×C₇×#41	1	2	yes	—
238	C₇	C₂×#2165	1	1	yes	characteristic, direct product factor, Sylow
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	C₇×#432	1	1	yes	characteristic
245	C₂²	C₇×#432	1	1	yes	characteristic
246	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	2	no	—
249	C₄	—	4	2	no	—
250	C₄	C₇×#432	1	1	yes	characteristic
251	C₄	C₇×#432	1	1	yes	characteristic
252	C₄	C₂×C₇×#117	1	1	yes	characteristic
253	C₄	C₂²×C₇×M₁₆	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	C₇×#2165	1	2	yes	direct product factor
258	C₂	C₂×C₇×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₇	#117
commutator	C₂×C₄	C₂³×C₄×C₇
Frattini, cosocle	C₂×C₈	C₂⁴×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
2. C₇×#2165	C₂	direct (N × H)
3. C₂×C₇×#364	C₂	semidirect
5. C₂×C₇×#347	C₂	semidirect
16. C₇×#364	C₂²	semidirect
18. C₇×#347	C₂²	semidirect
19. C₂×#2165	C₇	direct (N × H)
48. #2165	C₂×C₇	direct (N × H)
49. C₂×#364	C₂×C₇	semidirect
51. C₂×#347	C₂×C₇	semidirect
97. #364	C₂²×C₇	semidirect
99. #347	C₂²×C₇	semidirect
208. C₂×C₇	#2165	direct (N × H)
238. C₇	C₂×#2165	direct (N × H)
257. C₂	C₇×#2165	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
C₇×#2165	2
C₂×C₇×#364	2
C₂×C₇×#349	2
C₂×C₇×#347	2
C₂×#2165	7

Subgroups — C2×C7×#2165 #13210640

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#2165 #13210640