Subgroups — Dic₇×#364 #13269116

Order 1792.

398 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#364	C₂	1	1	yes	maximal, characteristic
2	Dic₇×M₃₂	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#89258	C₂	1	1	yes	maximal, characteristic
5	#11725616	C₂	1	2	yes	maximal
6	C₂×C₇×M₃₂	C₂²	1	2	yes	—
7	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₇	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₇	C₂²	1	1	yes	characteristic
12	C₂×#6386	C₂²	1	2	yes	—
13	C₄²×Dic₇	C₄	1	1	yes	characteristic
14	C₂×C₄×C₇⋊C₈	C₄	1	1	yes	characteristic
15	C₇×#364	C₄	1	4	yes	complemented
16	#89297	—	2	2	no	—
17	C₁₆×Dic₇	—	2	2	no	—
18	#89258	C₄	1	4	yes	complemented
19	C₄×#364	—	7	1	no	maximal, Sylow
20	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×Dic₇	—	2	2	no	—
24	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₇⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₇×M₃₂	—	2	2	no	—
29	C₇×M₃₂	C₂×C₄	1	4	yes	—
30	C₂×C₇×C₁₆	—	2	2	no	—
31	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
32	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
33	C₈×Dic₇	—	2	1	no	—
34	C₈×Dic₇	—	2	1	no	—
35	C₈×Dic₇	C₈	1	4	yes	—
36	C₈×Dic₇	C₈	1	4	yes	—
37	C₈×Dic₇	D₄	1	1	yes	characteristic
38	C₈×Dic₇	D₄	1	1	yes	characteristic
39	C₈×Dic₇	Q₈	1	1	yes	characteristic
40	C₈×Dic₇	Q₈	1	1	yes	characteristic
41	#6386	—	2	2	no	—
42	#6386	C₂×C₄	1	4	yes	—
43	C₂×C₇⋊C₁₆	—	2	2	no	—
44	C₄×C₇⋊C₈	—	2	2	no	—
45	C₂×#364	—	7	1	no	—
46	C₂×#364	D₇	1	1	yes	characteristic
47	C₄×M₃₂	—	7	2	no	—
48	C₄²×C₈	—	7	1	no	—
49	#1376	—	7	2	no	—
50	C₂²×C₄×C₇	—	2	1	no	—
51	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₇	—	2	1	no	—
53	C₂²×Dic₇	M₁₆	1	2	yes	—
54	C₂×C₇×C₈	C₄²	1	2	yes	—
55	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
56	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
57	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₇×C₈	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₇×C₈	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₇×C₈	#41	1	1	yes	characteristic
62	C₂×C₇×C₈	#41	1	1	yes	characteristic
63	C₄²×C₇	—	2	1	no	—
64	C₄²×C₇	C₄²	1	2	yes	—
65	C₄×Dic₇	—	2	8	no	—
66	C₄×Dic₇	—	2	1	no	—
67	C₄×Dic₇	C₄⋊C₄	1	1	yes	characteristic
68	C₄×Dic₇	C₄⋊C₄	1	1	yes	characteristic
69	C₄×Dic₇	C₂×C₈	1	2	yes	—
70	C₄×Dic₇	C₂×C₈	1	2	yes	—
71	C₂×C₇⋊C₈	—	2	1	no	—
72	C₂×C₇⋊C₈	—	2	1	no	—
73	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
74	C₂×C₇⋊C₈	C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
76	C₂×C₇⋊C₈	C₂×C₈	1	2	yes	—
77	C₂×C₇⋊C₈	M₁₆	1	2	yes	—
78	C₇×C₁₆	—	2	4	no	—
79	C₇⋊C₁₆	—	2	4	no	—
80	C₂×M₃₂	—	7	2	no	—
81	C₂×M₃₂	C₂×D₇	1	2	yes	—
82	C₂×C₄×C₈	—	7	1	no	—
83	C₂×C₄×C₈	—	7	1	no	—
84	C₂×C₄×C₈	—	7	1	no	—
85	C₂×C₄×C₈	—	7	1	no	—
86	C₂×C₄×C₈	—	7	1	no	—
87	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
88	C₄³	—	7	1	no	—
89	#364	—	7	4	no	—
90	#364	Dic₇	1	4	yes	direct product factor
91	#346	—	14	2	no	—
92	C₄×C₁₆	—	14	2	no	—
93	C₂³×C₇	C₂×M₁₆	1	1	yes	characteristic
94	C₂×C₄×C₇	—	2	4	no	—
95	C₂×C₄×C₇	—	2	2	no	—
96	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₇	#117	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
102	C₂×Dic₇	—	2	4	no	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	2	4	no	—
105	C₂×Dic₇	C₄⋊C₈	1	2	yes	—
106	C₂×Dic₇	C₄⋊C₈	1	2	yes	—
107	C₇×C₈	C₄×C₈	1	4	yes	—
108	C₇×C₈	C₄×D₄	1	1	yes	characteristic
109	C₇×C₈	C₄×D₄	1	1	yes	characteristic
110	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
111	C₇×C₈	C₄×Q₈	1	1	yes	characteristic
112	C₇⋊C₈	—	2	2	no	—
113	C₇⋊C₈	C₄⋊C₈	1	2	yes	—
114	C₇⋊C₈	C₄⋊C₈	1	2	yes	—
115	C₂²×C₈	—	7	1	no	—
116	C₂²×C₈	—	7	1	no	—
117	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
118	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
119	C₂×C₄²	—	7	1	no	—
120	C₂×C₄²	—	7	1	no	—
121	C₂×C₄²	—	14	2	no	—
122	C₂×C₄²	C₄×D₇	1	1	yes	characteristic
123	M₃₂	—	2	2	no	—
124	M₃₂	—	7	4	no	—
125	M₃₂	—	14	2	no	—
126	M₃₂	C₂×Dic₇	1	4	yes	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	14	2	no	—
129	C₄×C₈	—	7	4	no	—
130	C₄×C₈	—	7	4	no	—
131	C₄×C₈	—	7	1	no	—
132	C₄×C₈	—	7	1	no	—
133	C₄×C₈	—	7	1	no	—
134	C₄×C₈	—	7	1	no	—
135	C₄×C₈	—	14	2	no	—
136	C₄×C₈	—	14	1	no	—
137	C₄×C₈	—	14	1	no	—
138	C₄×C₈	C₄×D₇	1	2	yes	—
139	C₄×C₈	C₂×Dic₇	1	2	yes	—
140	C₂²×C₇	—	2	2	no	—
141	C₂²×C₇	C₄×M₁₆	1	1	yes	characteristic
142	C₂²×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂²×C₇	#431	1	1	yes	characteristic
144	C₄×C₇	—	2	4	no	—
145	C₄×C₇	C₄×C₄⋊C₄	1	1	yes	characteristic
146	C₄×C₇	C₂×C₄×C₈	1	1	yes	characteristic
147	C₄×C₇	C₂×C₄⋊C₈	1	1	yes	characteristic
148	C₄×C₇	#431	1	1	yes	characteristic
149	Dic₇	—	2	4	no	—
150	Dic₇	—	2	2	no	—
151	Dic₇	#364	1	4	yes	direct product factor
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	—	7	2	no	—
154	C₂²×C₄	—	14	1	no	—
155	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
156	C₂×C₈	—	7	1	no	—
157	C₂×C₈	—	7	1	no	—
158	C₂×C₈	—	7	2	no	—
159	C₂×C₈	—	7	2	no	—
160	C₂×C₈	—	7	2	no	—
161	C₂×C₈	—	14	1	no	—
162	C₂×C₈	—	14	1	no	—
163	C₂×C₈	C₈×D₇	1	2	yes	—
164	C₂×C₈	C₈×D₇	1	2	yes	—
165	C₂×C₈	C₄×Dic₇	1	2	yes	—
166	C₂×C₈	C₂×C₄×D₇	1	1	yes	characteristic
167	C₂×C₈	D₄×D₇	1	1	yes	characteristic
168	C₂×C₈	#1159	1	1	yes	characteristic
169	C₂×C₈	Q₈×D₇	1	1	yes	characteristic
170	C₂×C₈	#1161	1	1	yes	characteristic
171	C₂×C₈	C₂²×Dic₇	1	1	yes	characteristic
172	C₄²	—	2	1	no	—
173	C₄²	—	7	1	no	—
174	C₄²	—	7	1	no	—
175	C₄²	—	7	2	no	—
176	C₄²	—	7	2	no	—
177	C₄²	—	14	8	no	—
178	C₄²	—	14	1	no	—
179	C₄²	C₄×Dic₇	1	2	yes	—
180	C₁₆	—	2	4	no	—
181	C₁₆	—	14	4	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	—	2	1	no	—
184	C₂×C₇	C₄×C₄⋊C₈	1	1	yes	characteristic
185	C₂×C₇	C₂×#364	1	2	yes	—
186	C₂³	D₇×M₁₆	1	1	yes	characteristic
187	C₂×C₄	—	2	4	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	7	2	no	—
190	C₂×C₄	—	7	2	no	—
191	C₂×C₄	—	14	4	no	—
192	C₂×C₄	—	14	4	no	—
193	C₂×C₄	—	14	2	no	—
194	C₂×C₄	D₇×C₄⋊C₄	1	1	yes	characteristic
195	C₂×C₄	#6455	1	1	yes	characteristic
196	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₇	1	1	yes	characteristic
198	C₂×C₄	D₇×M₁₆	1	1	yes	characteristic
199	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
200	C₈	—	7	2	no	—
201	C₈	—	7	2	no	—
202	C₈	—	14	2	no	—
203	C₈	C₈×Dic₇	1	4	yes	—
204	C₈	#6450	1	1	yes	characteristic
205	C₈	#6456	1	1	yes	characteristic
206	C₈	D₄×Dic₇	1	1	yes	characteristic
207	C₈	Q₈×Dic₇	1	1	yes	characteristic
208	C₇	C₄×#364	1	1	yes	characteristic, complemented, Sylow
209	C₂²	—	2	2	no	—
210	C₂²	D₇×C₄⋊C₈	1	1	yes	characteristic
211	C₂²	#89605	1	1	yes	characteristic
212	C₂²	M₁₆×Dic₇	1	1	yes	characteristic
213	C₄	—	2	4	no	—
214	C₄	—	7	4	no	—
215	C₄	—	14	4	no	—
216	C₄	—	14	2	no	—
217	C₄	D₇×C₄⋊C₈	1	1	yes	characteristic
218	C₄	#89605	1	1	yes	characteristic
219	C₄	C₄⋊C₄×Dic₇	1	1	yes	characteristic
220	C₄	C₂×C₈×Dic₇	1	1	yes	characteristic
221	C₂	—	2	1	no	—
222	C₂	—	2	1	no	—
223	C₂	D₇×#364	1	2	yes	—
224	C₂	Dic₇×C₄⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇
commutator	C₄×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₇×#364	C₂
socle	C₂²×C₇	C₂×C₄⋊C₈
prime radical	C₂³×C₇	C₂×M₁₆
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×#364	D₇
hypocenter	C₇	C₄×#364
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
15. C₇×#364	C₄	semidirect
18. #89258	C₄	semidirect
90. #364	Dic₇	direct (N × H)
151. Dic₇	#364	direct (N × H)
208. C₇	C₄×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₇×#364	2
Dic₇×M₃₂	2
C₄×C₈×Dic₇	2
C₂×#89258	2
#11725616	2
C₄×#364	7

Subgroups — Dic7×#364 #13269116

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Dic₇×#364 #13269116