Subgroups — C₂×C₇×#924 #13061500

Order 1344.

644 conjugacy classes of subgroups.

The subgroup lattice has 208 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃×C₄×C₇	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₇×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₇×#217	C₂	1	2	yes	maximal
4	C₂×C₇×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₇×#924	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₇×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
7	C₂×C₇×C₂²⋊Q₈	—	3	1	no	maximal, Hall
8	C₂⁴×C₃×C₇	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₇	—	2	4	no	—
10	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₃×C₄×C₇	C₂²	1	4	yes	—
12	C₂²×C₃×C₄×C₇	C₂²	1	1	yes	characteristic
13	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
14	C₂²×C₇×Dic₃	C₂²	1	2	yes	—
15	C₂×C₇×Dic₆	—	2	4	no	—
16	C₂×C₇×Dic₆	C₂²	1	4	yes	complemented
17	C₇×#217	C₂²	1	8	yes	—
18	C₇×C₄⋊Dic₃	C₂²	1	4	yes	complemented
19	C₇×Dic₃⋊C₄	C₂²	1	8	yes	complemented
20	C₂²×C₇×Q₈	—	3	1	no	—
21	C₂³×C₄×C₇	D₃	1	1	yes	characteristic
22	C₇×C₂²⋊Q₈	—	3	8	no	—
23	C₂×C₇×C₄⋊C₄	—	3	2	no	—
24	C₂×C₇×C₄⋊C₄	—	3	1	no	—
25	C₂×C₇×C₂²⋊C₄	—	3	2	no	—
26	C₂×#924	C₇	1	1	yes	maximal, characteristic, direct product factor, Hall
27	C₂³×C₃×C₇	—	2	2	no	—
28	C₂³×C₃×C₇	—	2	2	no	—
29	C₂³×C₃×C₇	Q₈	1	4	yes	complemented
30	C₂³×C₃×C₇	C₂³	1	2	yes	—
31	C₂³×C₃×C₇	C₂³	1	1	yes	characteristic
32	C₂×C₃×C₄×C₇	—	2	2	no	—
33	C₂×C₃×C₄×C₇	—	2	8	no	—
34	C₂×C₃×C₄×C₇	D₄	1	4	yes	—
35	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
36	C₂×C₃×C₄×C₇	C₂³	1	2	yes	—
37	C₂×C₇×Dic₃	—	2	4	no	—
38	C₂×C₇×Dic₃	—	2	4	no	—
39	C₂×C₇×Dic₃	C₂³	1	4	yes	—
40	C₂×C₇×Dic₃	C₂³	1	4	yes	—
41	C₇×Dic₆	—	2	8	no	—
42	C₂⁴×C₇	C₂×D₃	1	1	yes	characteristic
43	C₂×C₇×Q₈	—	3	4	no	—
44	C₂×C₇×Q₈	—	6	4	no	—
45	C₂²×C₄×C₇	—	2	4	no	—
46	C₂²×C₄×C₇	—	3	2	no	—
47	C₂²×C₄×C₇	—	3	2	no	—
48	C₂²×C₄×C₇	C₂×D₃	1	4	yes	—
49	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×D₃	1	1	yes	characteristic
51	C₇×C₄⋊C₄	—	3	8	no	—
52	C₇×C₄⋊C₄	—	3	4	no	—
53	C₇×C₂²⋊C₄	—	3	8	no	—
54	C₂³×C₃×C₄	C₂×C₇	1	1	yes	characteristic
55	C₂²×Dic₆	C₂×C₇	1	1	yes	characteristic, complemented
56	C₂×#217	C₂×C₇	1	2	yes	—
57	C₂×C₄⋊Dic₃	C₂×C₇	1	1	yes	characteristic, complemented
58	#924	C₂×C₇	1	8	yes	direct product factor
59	C₂×Dic₃⋊C₄	C₂×C₇	1	2	yes	complemented
60	C₂²×C₃×C₇	—	2	8	no	—
61	C₂²×C₃×C₇	—	2	2	no	—
62	C₂²×C₃×C₇	—	2	2	no	—
63	C₂²×C₃×C₇	C₂×D₄	1	2	yes	—
64	C₂²×C₃×C₇	C₂×Q₈	1	2	yes	—
65	C₂²×C₃×C₇	C₂×Q₈	1	4	yes	complemented
66	C₂²×C₃×C₇	#41	1	2	yes	—
67	C₂²×C₃×C₇	C₂⁴	1	1	yes	characteristic
68	C₃×C₄×C₇	—	2	2	no	—
69	C₃×C₄×C₇	C₂×D₄	1	4	yes	—
70	C₇×Dic₃	—	2	4	no	—
71	C₇×Dic₃	—	2	4	no	—
72	C₂×C₂²⋊Q₈	—	3	1	no	Sylow
73	C₂³×C₇	—	2	2	no	—
74	C₂³×C₇	—	2	2	no	—
75	C₂³×C₇	Dic₆	1	4	yes	complemented
76	C₂³×C₇	C₂²×D₃	1	2	yes	—
77	C₂³×C₇	C₂²×D₃	1	1	yes	characteristic
78	C₇×Q₈	—	6	8	no	—
79	C₂×C₄×C₇	—	2	2	no	—
80	C₂×C₄×C₇	—	2	8	no	—
81	C₂×C₄×C₇	—	3	4	no	—
82	C₂×C₄×C₇	—	3	4	no	—
83	C₂×C₄×C₇	—	6	4	no	—
84	C₂×C₄×C₇	—	6	4	no	—
85	C₂×C₄×C₇	#67	1	4	yes	—
86	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
87	C₂×C₄×C₇	C₂²×D₃	1	2	yes	—
88	C₂⁴×C₃	C₂²×C₇	1	1	yes	characteristic
89	C₂²×C₃×C₄	—	2	4	no	—
90	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
91	C₂²×C₃×C₄	C₂²×C₇	1	4	yes	—
92	C₂²×C₃×C₄	C₂²×C₇	1	1	yes	characteristic
93	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
94	C₂²×Dic₃	C₂²×C₇	1	2	yes	—
95	C₂×Dic₆	—	2	4	no	—
96	C₂×Dic₆	C₂²×C₇	1	4	yes	complemented
97	#217	C₂²×C₇	1	8	yes	—
98	C₄⋊Dic₃	C₂²×C₇	1	4	yes	complemented
99	Dic₃⋊C₄	C₂²×C₇	1	8	yes	complemented
100	C₂×C₃×C₇	—	2	4	no	—
101	C₂×C₃×C₇	C₂²⋊Q₈	1	4	yes	complemented
102	C₂×C₃×C₇	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₃×C₇	C₂²×Q₈	1	1	yes	characteristic
104	C₂×C₃×C₇	C₂×#41	1	1	yes	characteristic
105	C₂²×Q₈	—	3	1	no	—
106	C₂³×C₄	D₃×C₇	1	1	yes	characteristic
107	C₂²⋊Q₈	—	3	8	no	—
108	C₂×C₄⋊C₄	—	3	2	no	—
109	C₂×C₄⋊C₄	—	3	1	no	—
110	C₂×C₂²⋊C₄	—	3	2	no	—
111	C₂²×C₇	—	2	8	no	—
112	C₂²×C₇	—	2	2	no	—
113	C₂²×C₇	—	2	2	no	—
114	C₂²×C₇	C₂×Dic₆	1	2	yes	—
115	C₂²×C₇	C₂×Dic₆	1	4	yes	complemented
116	C₂²×C₇	#235	1	2	yes	—
117	C₂²×C₇	C₂×#67	1	2	yes	—
118	C₂²×C₇	C₂³×D₃	1	1	yes	characteristic
119	C₄×C₇	—	2	2	no	—
120	C₄×C₇	—	6	4	no	—
121	C₄×C₇	—	6	4	no	—
122	C₄×C₇	C₂×#67	1	4	yes	—
123	C₂³×C₃	—	2	2	no	—
124	C₂³×C₃	—	2	2	no	—
125	C₂³×C₃	C₇×Q₈	1	4	yes	complemented
126	C₂³×C₃	C₂³×C₇	1	2	yes	—
127	C₂³×C₃	C₂³×C₇	1	1	yes	characteristic
128	C₂×C₃×C₄	—	2	2	no	—
129	C₂×C₃×C₄	—	2	8	no	—
130	C₂×C₃×C₄	C₇×D₄	1	4	yes	—
131	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
132	C₂×C₃×C₄	C₂³×C₇	1	2	yes	—
133	C₂×Dic₃	—	2	4	no	—
134	C₂×Dic₃	—	2	4	no	—
135	C₂×Dic₃	C₂³×C₇	1	4	yes	—
136	C₂×Dic₃	C₂³×C₇	1	4	yes	—
137	Dic₆	—	2	8	no	—
138	C₃×C₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Hall
139	C₂⁴	C₂×D₃×C₇	1	1	yes	characteristic
140	C₂×Q₈	—	3	4	no	—
141	C₂×Q₈	—	6	4	no	—
142	C₂²×C₄	—	2	4	no	—
143	C₂²×C₄	—	3	2	no	—
144	C₂²×C₄	—	3	2	no	—
145	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
146	C₂²×C₄	C₂×D₃×C₇	1	4	yes	—
147	C₂²×C₄	C₂×D₃×C₇	1	1	yes	characteristic
148	C₄⋊C₄	—	3	8	no	—
149	C₄⋊C₄	—	3	4	no	—
150	C₂²⋊C₄	—	3	8	no	—
151	C₂×C₇	—	2	4	no	—
152	C₂×C₇	#924	1	4	yes	direct product factor
153	C₂×C₇	C₂²×Dic₆	1	1	yes	characteristic
154	C₂×C₇	C₂×#235	1	1	yes	characteristic
155	C₂×C₇	C₂²×#67	1	1	yes	characteristic
156	C₂²×C₃	—	2	8	no	—
157	C₂²×C₃	—	2	2	no	—
158	C₂²×C₃	—	2	2	no	—
159	C₂²×C₃	C₂×C₇×D₄	1	2	yes	—
160	C₂²×C₃	C₂×C₇×Q₈	1	2	yes	—
161	C₂²×C₃	C₂×C₇×Q₈	1	4	yes	complemented
162	C₂²×C₃	C₇×#41	1	2	yes	—
163	C₂²×C₃	C₂⁴×C₇	1	1	yes	characteristic
164	C₃×C₄	—	2	2	no	—
165	C₃×C₄	C₂×C₇×D₄	1	4	yes	—
166	Dic₃	—	2	4	no	—
167	Dic₃	—	2	4	no	—
168	C₂³	—	2	2	no	—
169	C₂³	—	2	2	no	—
170	C₂³	C₇×Dic₆	1	4	yes	complemented
171	C₂³	C₂²×D₃×C₇	1	2	yes	—
172	C₂³	C₂²×D₃×C₇	1	1	yes	characteristic
173	Q₈	—	6	8	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	8	no	—
176	C₂×C₄	—	3	4	no	—
177	C₂×C₄	—	3	4	no	—
178	C₂×C₄	—	6	4	no	—
179	C₂×C₄	—	6	4	no	—
180	C₂×C₄	C₇×#67	1	4	yes	—
181	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
182	C₂×C₄	C₂²×D₃×C₇	1	2	yes	—
183	C₇	C₂×#924	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×C₃	—	2	4	no	—
185	C₂×C₃	C₇×C₂²⋊Q₈	1	4	yes	complemented
186	C₂×C₃	C₂²×C₇×D₄	1	1	yes	characteristic
187	C₂×C₃	C₂²×C₇×Q₈	1	1	yes	characteristic
188	C₂×C₃	C₂×C₇×#41	1	1	yes	characteristic
189	C₂²	—	2	8	no	—
190	C₂²	—	2	2	no	—
191	C₂²	—	2	2	no	—
192	C₂²	C₂×C₇×Dic₆	1	2	yes	—
193	C₂²	C₂×C₇×Dic₆	1	4	yes	complemented
194	C₂²	C₇×#235	1	2	yes	—
195	C₂²	C₂×C₇×#67	1	2	yes	—
196	C₂²	C₂³×D₃×C₇	1	1	yes	characteristic
197	C₄	—	2	2	no	—
198	C₄	—	6	4	no	—
199	C₄	—	6	4	no	—
200	C₄	C₂×C₇×#67	1	4	yes	—
201	C₃	C₂×C₇×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
202	C₂	—	2	4	no	—
203	C₂	C₇×#924	1	4	yes	direct product factor
204	C₂	C₂²×C₇×Dic₆	1	1	yes	characteristic
205	C₂	C₂×C₇×#235	1	1	yes	characteristic
206	C₂	C₂²×C₇×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₇	C₂²×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₇
Frattini	C₂²	C₂³×D₃×C₇
Fitting	C₂³×C₃×C₄×C₇	C₂
socle	C₂³×C₃×C₇	C₂³
prime radical	C₂⁴×C₃×C₇	C₂²
hypercenter	C₂³×C₄×C₇	D₃
hypocenter	C₃	C₂×C₇×C₂²⋊Q₈
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₇ ◁ C₂³×C₄×C₇

Complements

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

N	H ≅ G/N	type
2. C₂²×C₇×Dic₆	C₂	semidirect
4. C₂×C₇×C₄⋊Dic₃	C₂	semidirect
5. C₇×#924	C₂	direct (N × H)
6. C₂×C₇×Dic₃⋊C₄	C₂	semidirect
16. C₂×C₇×Dic₆	C₂²	semidirect
18. C₇×C₄⋊Dic₃	C₂²	semidirect
19. C₇×Dic₃⋊C₄	C₂²	semidirect
26. C₂×#924	C₇	direct (N × H)
29. C₂³×C₃×C₇	Q₈	semidirect
55. C₂²×Dic₆	C₂×C₇	semidirect
57. C₂×C₄⋊Dic₃	C₂×C₇	semidirect
58. #924	C₂×C₇	direct (N × H)
59. C₂×Dic₃⋊C₄	C₂×C₇	semidirect
65. C₂²×C₃×C₇	C₂×Q₈	semidirect
75. C₂³×C₇	Dic₆	semidirect
96. C₂×Dic₆	C₂²×C₇	semidirect
98. C₄⋊Dic₃	C₂²×C₇	semidirect
99. Dic₃⋊C₄	C₂²×C₇	semidirect
101. C₂×C₃×C₇	C₂²⋊Q₈	semidirect
115. C₂²×C₇	C₂×Dic₆	semidirect
125. C₂³×C₃	C₇×Q₈	semidirect
138. C₃×C₇	C₂×C₂²⋊Q₈	semidirect
152. C₂×C₇	#924	direct (N × H)
161. C₂²×C₃	C₂×C₇×Q₈	semidirect
170. C₂³	C₇×Dic₆	semidirect
183. C₇	C₂×#924	direct (N × H)
185. C₂×C₃	C₇×C₂²⋊Q₈	semidirect
193. C₂²	C₂×C₇×Dic₆	semidirect
201. C₃	C₂×C₇×C₂²⋊Q₈	semidirect
203. C₂	C₇×#924	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃×C₄×C₇	2
C₂²×C₇×Dic₆	2
C₂×C₇×#217	2
C₂×C₇×C₄⋊Dic₃	2
C₇×#924	2
C₂×C₇×Dic₃⋊C₄	2
C₂×C₇×C₂²⋊Q₈	3
C₂×#924	7

Subgroups — C2×C7×#924 #13061500

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇×#924 #13061500