Subgroups — C₂×#11731801 #13559893

Order 1792.

3774 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6451	C₂	1	6	yes	maximal
2	C₂×C₇×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×#89748	C₂	1	6	yes	maximal
4	C₂×#89707	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
6	#11731801	C₂	1	16	yes	maximal, direct product factor
7	C₂³×Dic₁₄	C₂²	1	4	yes	—
8	C₂²×C₇×C₄⋊C₄	C₂²	1	6	yes	—
9	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊Dic₇	C₂²	1	6	yes	—
11	C₂²×Dic₇⋊C₄	C₂²	1	12	yes	—
12	C₂²×C₄×Dic₇	C₂²	1	3	yes	—
13	C₂²×C₄×Dic₇	C₂²	1	3	yes	—
14	C₂×#6451	—	2	48	no	—
15	C₂×#6451	C₂²	1	48	yes	—
16	C₇×#384	C₂²	1	8	yes	—
17	#89748	C₂²	1	48	yes	—
18	#89707	C₂²	1	8	yes	—
19	C₄²×Dic₇	C₂²	1	8	yes	—
20	C₂×#2868	—	7	1	no	maximal, Sylow
21	C₂³×C₄×C₇	C₂³	1	4	yes	—
22	C₂³×C₄×C₇	C₂³	1	3	yes	—
23	C₂³×Dic₇	C₂³	1	4	yes	—
24	C₂³×Dic₇	C₂³	1	1	yes	characteristic
25	C₂³×Dic₇	C₂³	1	3	yes	—
26	C₂²×Dic₁₄	—	2	48	no	—
27	C₂²×Dic₁₄	C₂³	1	16	yes	—
28	C₂×C₇×C₄⋊C₄	—	2	24	no	—
29	C₂×C₇×C₄⋊C₄	C₂³	1	24	yes	—
30	C₂×C₄²×C₇	Q₈	1	8	yes	complemented
31	C₂×C₄²×C₇	C₂³	1	4	yes	—
32	C₂×C₄⋊Dic₇	—	2	24	no	—
33	C₂×C₄⋊Dic₇	C₂³	1	24	yes	—
34	C₂×Dic₇⋊C₄	—	2	48	no	—
35	C₂×Dic₇⋊C₄	C₂³	1	48	yes	—
36	C₂×C₄×Dic₇	D₄	1	24	yes	—
37	C₂×C₄×Dic₇	Q₈	1	24	yes	complemented
38	C₂×C₄×Dic₇	C₂³	1	12	yes	—
39	C₂×C₄×Dic₇	C₂³	1	12	yes	—
40	#6451	—	2	192	no	—
41	C₂²×C₄⋊Q₈	—	7	6	no	—
42	#2868	—	7	16	no	—
43	C₂×#384	—	7	1	no	—
44	C₂×#384	—	7	6	no	—
45	C₂×#384	D₇	1	1	yes	characteristic
46	C₂×C₄³	—	7	1	no	—
47	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
48	C₂²×C₄×C₇	—	2	24	no	—
49	C₂²×C₄×C₇	C₂×D₄	1	12	yes	—
50	C₂²×C₄×C₇	C₂×Q₈	1	12	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	12	yes	—
52	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
53	C₂²×C₄×C₇	C₂⁴	1	6	yes	—
54	C₂²×Dic₇	—	2	24	no	—
55	C₂²×Dic₇	C₂×D₄	1	12	yes	—
56	C₂²×Dic₇	C₂×D₄	1	12	yes	—
57	C₂²×Dic₇	C₂×Q₈	1	24	yes	—
58	C₂²×Dic₇	C₂⁴	1	8	yes	—
59	C₂²×Dic₇	C₂⁴	1	2	yes	—
60	C₂²×Dic₇	C₂⁴	1	6	yes	—
61	C₂×Dic₁₄	—	2	96	no	—
62	C₂×Dic₁₄	—	4	64	no	—
63	C₇×C₄⋊C₄	—	2	48	no	—
64	C₄²×C₇	C₂×Q₈	1	16	yes	complemented
65	C₄⋊Dic₇	—	2	48	no	—
66	Dic₇⋊C₄	—	2	96	no	—
67	C₄×Dic₇	C₂×D₄	1	48	yes	—
68	C₄×Dic₇	C₂×Q₈	1	48	yes	complemented
69	C₂³×Q₈	—	7	4	no	—
70	C₂×C₄⋊Q₈	—	7	48	no	—
71	C₂×C₄⋊Q₈	—	14	48	no	—
72	C₂²×C₄⋊C₄	—	7	12	no	—
73	C₂²×C₄⋊C₄	—	7	6	no	—
74	C₂²×C₄⋊C₄	C₂×D₇	1	6	yes	—
75	C₂²×C₄²	—	7	3	no	—
76	C₂²×C₄²	—	7	3	no	—
77	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
78	#384	—	7	48	no	—
79	#384	—	7	8	no	—
80	#384	C₂×D₇	1	8	yes	—
81	C₄³	—	7	8	no	—
82	C₂³×C₇	C₂²×D₄	1	6	yes	—
83	C₂³×C₇	C₂²×Q₈	1	2	yes	—
84	C₂³×C₇	C₂²×Q₈	1	6	yes	—
85	C₂³×C₇	C₂⁵	1	1	yes	characteristic
86	C₂×C₄×C₇	—	2	24	no	—
87	C₂×C₄×C₇	—	4	16	no	—
88	C₂×C₄×C₇	C₄⋊Q₈	1	48	yes	complemented
89	C₂×C₄×C₇	C₂²×D₄	1	12	yes	—
90	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
91	C₂×C₄×C₇	C₂²×Q₈	1	12	yes	—
92	C₂×Dic₇	—	2	24	no	—
93	C₂×Dic₇	—	4	16	no	—
94	C₂×Dic₇	#127	1	16	yes	—
95	C₂×Dic₇	C₄⋊Q₈	1	48	yes	complemented
96	C₂×Dic₇	C₂²×D₄	1	12	yes	—
97	C₂×Dic₇	C₂²×D₄	1	12	yes	—
98	C₂×Dic₇	C₂²×Q₈	1	24	yes	—
99	Dic₁₄	—	4	64	no	—
100	C₂²×Q₈	—	7	16	no	—
101	C₂²×Q₈	—	14	48	no	—
102	C₂³×C₄	—	7	3	no	—
103	C₂³×C₄	—	7	1	no	—
104	C₂³×C₄	—	7	4	no	—
105	C₂³×C₄	C₂²×D₇	1	4	yes	—
106	C₂³×C₄	C₂²×D₇	1	3	yes	—
107	C₄⋊Q₈	—	14	192	no	—
108	C₂×C₄⋊C₄	—	2	24	no	—
109	C₂×C₄⋊C₄	—	7	48	no	—
110	C₂×C₄⋊C₄	—	7	24	no	—
111	C₂×C₄⋊C₄	—	14	48	no	—
112	C₂×C₄⋊C₄	—	14	24	no	—
113	C₂×C₄⋊C₄	C₂²×D₇	1	24	yes	—
114	C₂×C₄²	—	7	24	no	—
115	C₂×C₄²	—	7	24	no	—
116	C₂×C₄²	—	7	12	no	—
117	C₂×C₄²	—	7	12	no	—
118	C₂×C₄²	Dic₁₄	1	8	yes	complemented
119	C₂×C₄²	C₂²×D₇	1	4	yes	—
120	C₂²×C₇	C₂×#127	1	4	yes	—
121	C₂²×C₇	C₂×C₄⋊Q₈	1	12	yes	—
122	C₂²×C₇	C₂×C₄⋊Q₈	1	12	yes	—
123	C₂²×C₇	C₂³×D₄	1	3	yes	—
124	C₂²×C₇	C₂³×Q₈	1	1	yes	characteristic
125	C₂²×C₇	C₂³×Q₈	1	3	yes	—
126	C₄×C₇	—	4	8	no	—
127	C₄×C₇	C₂×C₄⋊Q₈	1	24	yes	complemented
128	Dic₇	—	4	8	no	—
129	Dic₇	C₂×#127	1	8	yes	—
130	Dic₇	C₂×C₄⋊Q₈	1	24	yes	complemented
131	C₂⁴	C₂³×D₇	1	1	yes	characteristic
132	C₂×Q₈	—	14	96	no	—
133	C₂×Q₈	—	28	64	no	—
134	C₂²×C₄	—	2	24	no	—
135	C₂²×C₄	—	7	12	no	—
136	C₂²×C₄	—	7	12	no	—
137	C₂²×C₄	—	7	24	no	—
138	C₂²×C₄	—	7	6	no	—
139	C₂²×C₄	—	7	2	no	—
140	C₂²×C₄	—	7	8	no	—
141	C₂²×C₄	—	14	24	no	—
142	C₂²×C₄	C₂×Dic₁₄	1	12	yes	—
143	C₂²×C₄	D₄×D₇	1	12	yes	—
144	C₂²×C₄	Q₈×D₇	1	12	yes	—
145	C₂²×C₄	C₂³×D₇	1	8	yes	—
146	C₂²×C₄	C₂³×D₇	1	6	yes	—
147	C₄⋊C₄	—	2	48	no	—
148	C₄⋊C₄	—	14	96	no	—
149	C₄⋊C₄	—	14	48	no	—
150	C₄²	—	7	48	no	—
151	C₄²	—	7	48	no	—
152	C₄²	C₂×Dic₁₄	1	16	yes	complemented
153	C₂×C₇	#2868	1	8	yes	complemented
154	C₂×C₇	C₂²×#127	1	1	yes	characteristic
155	C₂×C₇	C₂²×C₄⋊Q₈	1	3	yes	—
156	C₂×C₇	C₂²×C₄⋊Q₈	1	3	yes	—
157	C₂³	C₂²×Dic₁₄	1	2	yes	—
158	C₂³	C₂×D₄×D₇	1	6	yes	—
159	C₂³	C₂×Q₈×D₇	1	6	yes	—
160	C₂³	C₂⁴×D₇	1	1	yes	characteristic
161	Q₈	—	28	64	no	—
162	C₂×C₄	—	2	24	no	—
163	C₂×C₄	—	4	16	no	—
164	C₂×C₄	—	7	16	no	—
165	C₂×C₄	—	7	48	no	—
166	C₂×C₄	—	7	12	no	—
167	C₂×C₄	—	7	12	no	—
168	C₂×C₄	—	7	24	no	—
169	C₂×C₄	—	14	24	no	—
170	C₂×C₄	—	28	16	no	—
171	C₂×C₄	#6451	1	48	yes	complemented
172	C₂×C₄	C₂²×Dic₁₄	1	12	yes	—
173	C₂×C₄	C₂×D₄×D₇	1	12	yes	—
174	C₂×C₄	C₂×Q₈×D₇	1	12	yes	—
175	C₇	C₂×#2868	1	1	yes	characteristic, complemented, Sylow
176	C₂²	C₂×#6451	1	12	yes	—
177	C₂²	D₇×#127	1	4	yes	—
178	C₂²	D₇×C₄⋊Q₈	1	12	yes	—
179	C₂²	C₂³×Dic₁₄	1	1	yes	characteristic
180	C₂²	C₂²×D₄×D₇	1	3	yes	—
181	C₂²	C₂²×Q₈×D₇	1	3	yes	—
182	C₄	—	4	8	no	—
183	C₄	—	7	8	no	—
184	C₄	—	7	24	no	—
185	C₄	—	28	8	no	—
186	C₄	C₂×#6451	1	24	yes	complemented
187	C₂	#11731801	1	8	yes	direct product factor
188	C₂	C₂²×#6451	1	3	yes	—
189	C₂	C₂×D₇×#127	1	1	yes	characteristic
190	C₂	C₂×D₇×C₄⋊Q₈	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator, cosocle	C₂³×C₇	C₂⁵
Frattini	C₂³	C₂⁴×D₇
Fitting	C₂×C₇×#384	C₂
socle, prime radical	C₂⁴×C₇	C₂⁴
hypercenter	C₂×#384	D₇
hypocenter	C₇	C₂×#2868
LCS	G ▷ C₂³×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
6. #11731801	C₂	direct (N × H)
30. C₂×C₄²×C₇	Q₈	semidirect
37. C₂×C₄×Dic₇	Q₈	semidirect
64. C₄²×C₇	C₂×Q₈	semidirect
68. C₄×Dic₇	C₂×Q₈	semidirect
88. C₂×C₄×C₇	C₄⋊Q₈	semidirect
95. C₂×Dic₇	C₄⋊Q₈	semidirect
118. C₂×C₄²	Dic₁₄	semidirect
127. C₄×C₇	C₂×C₄⋊Q₈	semidirect
130. Dic₇	C₂×C₄⋊Q₈	semidirect
152. C₄²	C₂×Dic₁₄	semidirect
153. C₂×C₇	#2868	semidirect
171. C₂×C₄	#6451	semidirect
175. C₇	C₂×#2868	semidirect
186. C₄	C₂×#6451	semidirect
187. C₂	#11731801	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#6451	2
C₂×C₇×#384	2
C₂×#89748	2
C₂×#89707	2
C₂×C₄²×Dic₇	2
#11731801	2
C₂×#2868	7

Subgroups — C2×#11731801 #13559893

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11731801 #13559893