Subgroups — C₇×#23837 #13201053

Order 1792.

778 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₇×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₇×#2084	C₂	1	4	yes	maximal
3	C₇×#2058	C₂	1	4	yes	maximal
4	C₄×C₇×#106	C₂	1	4	yes	maximal
5	C₇×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₇×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×C₇×Q₈	—	2	2	no	—
8	C₂×C₇×#106	C₂²	1	8	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	2	yes	—
10	C₇×#391	—	2	2	no	—
11	C₇×#389	—	2	2	no	—
12	C₇×#389	C₂²	1	4	yes	—
13	C₇×#384	C₂²	1	2	yes	—
14	C₄×C₇×C₄⋊C₄	—	2	2	no	—
15	C₄×C₇×C₄⋊C₄	C₂²	1	4	yes	—
16	C₄³×C₇	C₂²	1	1	yes	characteristic
17	C₇×#340	C₂²	1	4	yes	—
18	C₇×#340	C₂²	1	4	yes	—
19	C₇×#336	C₂²	1	4	yes	—
20	#23837	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	—	2	2	no	—
22	C₂²×C₇×C₈	C₂³	1	4	yes	—
23	C₇×C₄⋊Q₈	—	2	8	no	—
24	C₄×C₇×Q₈	—	4	8	no	—
25	C₂×C₇×C₄⋊C₄	—	2	2	no	—
26	C₂×C₇×C₄⋊C₄	—	2	2	no	—
27	C₂×C₇×C₄⋊C₄	—	2	2	no	—
28	C₂×C₇×C₄⋊C₄	—	2	4	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₇×C₄⋊C₄	C₂³	1	4	yes	—
31	C₂×C₄²×C₇	—	2	4	no	—
32	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₇	D₄	1	2	yes	—
35	C₂×C₄²×C₇	C₂³	1	2	yes	—
36	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
37	C₇×#106	—	2	16	no	—
38	C₄×C₇×C₈	Q₈	1	8	yes	complemented
39	C₇×#95	—	2	4	no	—
40	C₇×#95	—	2	4	no	—
41	C₄×C₄⋊Q₈	C₂×C₇	1	2	yes	—
42	#2084	C₂×C₇	1	4	yes	—
43	#2058	C₂×C₇	1	4	yes	—
44	C₄×#106	C₂×C₇	1	4	yes	—
45	#1744	C₂×C₇	1	1	yes	characteristic
46	C₂×C₇×Q₈	—	2	8	no	—
47	C₂×C₇×Q₈	—	4	4	no	—
48	C₂×C₇×Q₈	—	4	4	no	—
49	C₂²×C₄×C₇	—	2	4	no	—
50	C₂²×C₄×C₇	—	2	4	no	—
51	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
56	C₂×C₇×C₈	—	2	8	no	—
57	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
58	C₇×C₄⋊C₄	—	2	8	no	—
59	C₇×C₄⋊C₄	—	2	4	no	—
60	C₇×C₄⋊C₄	—	2	8	no	—
61	C₇×C₄⋊C₄	—	4	4	no	—
62	C₇×C₄⋊C₄	—	4	8	no	—
63	C₇×C₄⋊C₄	—	4	4	no	—
64	C₇×C₄⋊C₄	#41	1	8	yes	—
65	C₄²×C₇	—	2	4	no	—
66	C₄²×C₇	—	4	1	no	—
67	C₄²×C₇	—	4	1	no	—
68	C₄²×C₇	—	4	8	no	—
69	C₄²×C₇	SD₁₆	1	8	yes	—
70	C₄²×C₇	C₂×Q₈	1	4	yes	—
71	C₂×C₄⋊Q₈	C₂²×C₇	1	2	yes	—
72	C₂×C₄×Q₈	—	2	2	no	—
73	C₂×#106	C₂²×C₇	1	8	yes	—
74	C₂×C₄×C₈	C₂²×C₇	1	2	yes	—
75	#391	—	2	2	no	—
76	#389	—	2	2	no	—
77	#389	C₂²×C₇	1	4	yes	—
78	#384	C₂²×C₇	1	2	yes	—
79	C₄×C₄⋊C₄	—	2	2	no	—
80	C₄×C₄⋊C₄	C₂²×C₇	1	4	yes	—
81	C₄³	C₂²×C₇	1	1	yes	characteristic
82	#340	C₂²×C₇	1	4	yes	—
83	#340	C₂²×C₇	1	4	yes	—
84	#336	C₂²×C₇	1	4	yes	—
85	C₂³×C₇	C₂²×D₄	1	1	yes	characteristic
86	C₇×Q₈	—	4	16	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	—	2	2	no	—
90	C₂×C₄×C₇	—	2	1	no	—
91	C₂×C₄×C₇	—	2	1	no	—
92	C₂×C₄×C₇	—	2	2	no	—
93	C₂×C₄×C₇	—	2	8	no	—
94	C₂×C₄×C₇	—	4	4	no	—
95	C₂×C₄×C₇	—	4	4	no	—
96	C₂×C₄×C₇	—	4	4	no	—
97	C₂×C₄×C₇	—	4	4	no	—
98	C₂×C₄×C₇	—	4	4	no	—
99	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
100	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
101	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
103	C₂×C₄×C₇	C₂×SD₁₆	1	4	yes	—
104	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
105	C₂×C₄×C₇	C₂×#41	1	4	yes	—
106	C₇×C₈	—	2	8	no	—
107	C₂²×Q₈	—	2	2	no	—
108	C₂²×C₈	C₂³×C₇	1	4	yes	—
109	C₄⋊Q₈	—	2	8	no	—
110	C₄×Q₈	—	4	8	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	2	2	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
116	C₂×C₄⋊C₄	C₂³×C₇	1	4	yes	—
117	C₂×C₄²	—	2	4	no	—
118	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
119	C₂×C₄²	C₇×D₄	1	1	yes	characteristic
120	C₂×C₄²	C₇×D₄	1	2	yes	—
121	C₂×C₄²	C₂³×C₇	1	2	yes	—
122	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
123	#106	—	2	16	no	—
124	C₄×C₈	C₇×Q₈	1	8	yes	complemented
125	#95	—	2	4	no	—
126	#95	—	2	4	no	—
127	C₂²×C₇	#525	1	2	yes	—
128	C₂²×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
129	C₂²×C₇	C₂²×SD₁₆	1	2	yes	—
130	C₂²×C₇	#575	1	1	yes	characteristic
131	C₂²×C₇	#575	1	1	yes	characteristic
132	C₄×C₇	—	2	4	no	—
133	C₄×C₇	—	2	4	no	—
134	C₄×C₇	—	2	2	no	—
135	C₄×C₇	—	4	8	no	—
136	C₄×C₇	—	8	4	no	—
137	C₄×C₇	#498	1	4	yes	—
138	C₄×C₇	#533	1	4	yes	—
139	C₂×Q₈	—	2	8	no	—
140	C₂×Q₈	—	4	4	no	—
141	C₂×Q₈	—	4	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
145	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
146	C₂²×C₄	C₂×C₇×D₄	1	2	yes	—
147	C₂²×C₄	C₂×C₇×D₄	1	1	yes	characteristic
148	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
149	C₂×C₈	—	2	8	no	—
150	C₂×C₈	C₂×C₇×Q₈	1	8	yes	—
151	C₄⋊C₄	—	2	8	no	—
152	C₄⋊C₄	—	2	4	no	—
153	C₄⋊C₄	—	2	8	no	—
154	C₄⋊C₄	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	4	no	—
157	C₄⋊C₄	C₇×#41	1	8	yes	—
158	C₄²	—	2	4	no	—
159	C₄²	—	4	1	no	—
160	C₄²	—	4	1	no	—
161	C₄²	—	4	8	no	—
162	C₄²	C₇×SD₁₆	1	8	yes	—
163	C₄²	C₂×C₇×Q₈	1	4	yes	—
164	C₂×C₇	#2399	1	1	yes	characteristic
165	C₂×C₇	#3101	1	2	yes	—
166	C₂×C₇	#3103	1	2	yes	—
167	C₂×C₇	C₂×#498	1	1	yes	characteristic
168	C₂×C₇	#3162	1	1	yes	characteristic
169	C₂³	C₂²×C₇×D₄	1	1	yes	characteristic
170	Q₈	—	4	16	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	1	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	8	no	—
178	C₂×C₄	—	4	4	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
184	C₂×C₄	C₇×C₄⋊Q₈	1	2	yes	—
185	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
186	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
187	C₂×C₄	C₂×C₇×SD₁₆	1	4	yes	—
188	C₂×C₄	C₂²×C₇×Q₈	1	2	yes	—
189	C₂×C₄	C₂×C₇×#41	1	4	yes	—
190	C₈	—	2	8	no	—
191	C₇	#23837	1	1	yes	characteristic, direct product factor, Sylow
192	C₂²	C₇×#525	1	2	yes	—
193	C₂²	C₂×C₇×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²	C₂²×C₇×SD₁₆	1	2	yes	—
195	C₂²	C₇×#575	1	1	yes	characteristic
196	C₂²	C₇×#575	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	4	no	—
200	C₄	—	4	8	no	—
201	C₄	—	8	4	no	—
202	C₄	C₇×#498	1	4	yes	—
203	C₄	C₇×#533	1	4	yes	—
204	C₂	C₇×#2399	1	1	yes	characteristic
205	C₂	C₇×#3101	1	2	yes	—
206	C₂	C₇×#3103	1	2	yes	—
207	C₂	C₂×C₇×#498	1	1	yes	characteristic
208	C₂	C₇×#3162	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ C₄³×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. #23837	C₇	direct (N × H)
38. C₄×C₇×C₈	Q₈	semidirect
124. C₄×C₈	C₇×Q₈	semidirect
191. C₇	#23837	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₇×C₄⋊Q₈	2
C₇×#2084	2
C₇×#2058	2
C₄×C₇×#106	2
C₇×#1744	2
#23837	7

Subgroups — C7×#23837 #13201053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×#23837 #13201053