Subgroups — C₂×#11740676 #13516431

Order 1792.

2326 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#90002	C₂	1	4	yes	maximal
3	#11740676	C₂	1	16	yes	maximal, direct product factor
4	C₂×#89753	C₂	1	8	yes	maximal
5	C₂×#89709	C₂	1	2	yes	maximal
6	C₂³×C₇×Q₈	C₂²	1	2	yes	—
7	C₂×C₇×C₄⋊Q₈	C₄	1	16	yes	complemented
8	C₂×C₇×C₄⋊Q₈	C₂²	1	8	yes	—
9	C₂²×C₇×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₇	C₂²	1	4	yes	—
12	C₂²×C₄×Dic₇	C₂²	1	4	yes	—
13	#90002	C₂²	1	32	yes	—
14	C₂×C₇⋊#95	C₂²	1	4	yes	—
15	C₂×C₇⋊#95	C₂²	1	8	yes	—
16	C₂×C₇⋊#95	C₂²	1	8	yes	—
17	#89753	C₂²	1	64	yes	—
18	#89709	C₂²	1	16	yes	—
19	C₂×#2381	—	7	1	no	maximal, Sylow
20	C₂²×C₇×Q₈	—	4	4	no	—
21	C₂²×C₇×Q₈	—	4	4	no	—
22	C₂²×C₇×Q₈	C₂×C₄	1	16	yes	—
23	C₂²×C₇×Q₈	C₂³	1	8	yes	—
24	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₇	C₂³	1	4	yes	—
26	C₂³×C₄×C₇	C₂³	1	2	yes	—
27	C₂³×Dic₇	C₂³	1	8	yes	—
28	C₇×C₄⋊Q₈	C₂×C₄	1	64	yes	complemented
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	32	yes	—
30	C₂×C₇×C₄⋊C₄	C₂³	1	16	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	8	yes	—
32	C₂×C₄²×C₇	C₂³	1	4	yes	—
33	C₂×C₄⋊Dic₇	—	4	8	no	—
34	C₂×C₄⋊Dic₇	C₂³	1	16	yes	—
35	C₂×C₄×Dic₇	—	4	8	no	—
36	C₂×C₄×Dic₇	C₂³	1	16	yes	—
37	C₇⋊#95	C₂³	1	16	yes	—
38	C₇⋊#95	C₂³	1	32	yes	—
39	C₇⋊#95	C₂³	1	32	yes	—
40	C₂²×C₄⋊Q₈	D₇	1	1	yes	characteristic
41	#2381	—	7	16	no	—
42	C₂×#391	—	7	4	no	—
43	C₂×#387	—	7	8	no	—
44	C₂×#383	—	7	2	no	—
45	C₂⁴×C₇	C₂⁴	1	1	yes	characteristic
46	C₂×C₇×Q₈	—	4	32	no	—
47	C₂×C₇×Q₈	—	4	8	no	—
48	C₂×C₇×Q₈	—	4	8	no	—
49	C₂×C₇×Q₈	C₂²×C₄	1	32	yes	—
50	C₂²×C₄×C₇	—	2	8	no	—
51	C₂²×C₄×C₇	—	2	8	no	—
52	C₂²×C₄×C₇	—	2	4	no	—
53	C₂²×C₄×C₇	—	2	8	no	—
54	C₂²×C₄×C₇	C₂²×C₄	1	8	yes	—
55	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₇	C₂⁴	1	4	yes	—
58	C₂²×C₄×C₇	C₂⁴	1	2	yes	—
59	C₂²×C₄×C₇	C₂⁴	1	8	yes	—
60	C₂²×Dic₇	—	2	16	no	—
61	C₂²×Dic₇	—	2	32	no	—
62	C₂²×Dic₇	C₂⁴	1	16	yes	—
63	C₇×C₄⋊C₄	C₂²×C₄	1	64	yes	—
64	C₄²×C₇	C₂²×C₄	1	16	yes	—
65	C₄⋊Dic₇	—	4	16	no	—
66	C₄×Dic₇	—	4	16	no	—
67	C₂³×Q₈	C₂×D₇	1	2	yes	—
68	C₂×C₄⋊Q₈	Dic₇	1	16	yes	complemented
69	C₂×C₄⋊Q₈	C₂×D₇	1	8	yes	—
70	C₂²×C₄⋊C₄	—	7	4	no	—
71	C₂²×C₄⋊C₄	C₂×D₇	1	4	yes	—
72	C₂²×C₄²	—	7	4	no	—
73	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
74	#391	—	7	32	no	—
75	#387	—	7	64	no	—
76	#383	—	7	16	no	—
77	C₂×#95	—	7	8	no	—
78	C₂×#95	—	7	4	no	—
79	C₂×#95	—	7	8	no	—
80	C₂³×C₇	C₂³×C₄	1	2	yes	—
81	C₂³×C₇	H(5,𝔽₂)	1	4	yes	—
82	C₂³×C₇	2₋¹⁺⁴	1	8	yes	—
83	C₂³×C₇	C₂⁵	1	1	yes	characteristic
84	C₇×Q₈	—	4	32	no	—
85	C₂×C₄×C₇	—	2	32	no	—
86	C₂×C₄×C₇	—	2	8	no	—
87	C₂×C₄×C₇	—	2	16	no	—
88	C₂×C₄×C₇	—	2	8	no	—
89	C₂×C₄×C₇	—	2	8	no	—
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	2	8	no	—
92	C₂×C₄×C₇	C₂³×C₄	1	8	yes	—
93	C₂×C₄×C₇	C₂³×C₄	1	4	yes	—
94	C₂×C₄×C₇	C₂³×C₄	1	16	yes	—
95	C₂×Dic₇	—	2	16	no	—
96	C₂×Dic₇	—	2	32	no	—
97	C₂×Dic₇	—	4	32	no	—
98	C₂²×Q₈	—	4	4	no	—
99	C₂²×Q₈	—	4	4	no	—
100	C₂²×Q₈	C₂×Dic₇	1	16	yes	—
101	C₂²×Q₈	C₂²×D₇	1	8	yes	—
102	C₂³×C₄	—	7	8	no	—
103	C₂³×C₄	C₂²×D₇	1	4	yes	—
104	C₂³×C₄	C₂²×D₇	1	2	yes	—
105	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
106	C₄⋊Q₈	C₂×Dic₇	1	64	yes	complemented
107	C₂×C₄⋊C₄	—	7	16	no	—
108	C₂×C₄⋊C₄	—	28	8	no	—
109	C₂×C₄⋊C₄	C₂×Dic₇	1	32	yes	—
110	C₂×C₄⋊C₄	C₂²×D₇	1	16	yes	—
111	C₂×C₄²	—	7	16	no	—
112	C₂×C₄²	—	28	8	no	—
113	C₂×C₄²	C₂×Dic₇	1	8	yes	—
114	C₂×C₄²	C₂²×D₇	1	4	yes	—
115	#95	—	7	32	no	—
116	#95	—	7	16	no	—
117	#95	—	7	32	no	—
118	C₂²×C₇	#518	1	4	yes	—
119	C₂²×C₇	#519	1	8	yes	—
120	C₂²×C₇	#563	1	16	yes	—
121	C₂²×C₇	C₂⁴×C₄	1	1	yes	characteristic
122	C₂²×C₇	C₂×H(5,𝔽₂)	1	2	yes	—
123	C₂²×C₇	C₂×2₋¹⁺⁴	1	4	yes	—
124	C₄×C₇	—	2	16	no	—
125	C₄×C₇	—	2	4	no	—
126	C₄×C₇	—	2	8	no	—
127	Dic₇	—	4	16	no	—
128	C₂⁴	C₂³×D₇	1	1	yes	characteristic
129	C₂×Q₈	—	4	32	no	—
130	C₂×Q₈	—	4	8	no	—
131	C₂×Q₈	—	4	8	no	—
132	C₂×Q₈	C₂²×Dic₇	1	32	yes	—
133	C₂²×C₄	—	2	8	no	—
134	C₂²×C₄	—	2	8	no	—
135	C₂²×C₄	—	2	4	no	—
136	C₂²×C₄	—	2	8	no	—
137	C₂²×C₄	—	7	16	no	—
138	C₂²×C₄	—	14	16	no	—
139	C₂²×C₄	—	14	32	no	—
140	C₂²×C₄	C₂²×Dic₇	1	8	yes	—
141	C₂²×C₄	C₂²×Dic₇	1	4	yes	—
142	C₂²×C₄	C₂²×Dic₇	1	16	yes	—
143	C₂²×C₄	C₂³×D₇	1	4	yes	—
144	C₂²×C₄	C₂³×D₇	1	2	yes	—
145	C₂²×C₄	C₂³×D₇	1	8	yes	—
146	C₄⋊C₄	—	28	16	no	—
147	C₄⋊C₄	C₂²×Dic₇	1	64	yes	—
148	C₄²	—	28	16	no	—
149	C₄²	C₂²×Dic₇	1	16	yes	—
150	C₂×C₇	#2381	1	8	yes	complemented
151	C₂×C₇	C₂×#518	1	1	yes	characteristic
152	C₂×C₇	C₂×#519	1	2	yes	—
153	C₂×C₇	C₂×#563	1	4	yes	—
154	C₂³	#6548	1	4	yes	—
155	C₂³	#6551	1	8	yes	—
156	C₂³	C₂³×Dic₇	1	2	yes	—
157	C₂³	C₂⁴×D₇	1	1	yes	characteristic
158	Q₈	—	4	32	no	—
159	C₂×C₄	—	2	32	no	—
160	C₂×C₄	—	2	8	no	—
161	C₂×C₄	—	2	16	no	—
162	C₂×C₄	—	2	8	no	—
163	C₂×C₄	—	2	8	no	—
164	C₂×C₄	—	2	4	no	—
165	C₂×C₄	—	2	8	no	—
166	C₂×C₄	—	14	16	no	—
167	C₂×C₄	—	14	32	no	—
168	C₂×C₄	—	28	32	no	—
169	C₂×C₄	C₂³×Dic₇	1	8	yes	—
170	C₂×C₄	C₂³×Dic₇	1	4	yes	—
171	C₂×C₄	C₂³×Dic₇	1	16	yes	—
172	C₇	C₂×#2381	1	1	yes	characteristic, complemented, Sylow
173	C₂²	#90426	1	16	yes	—
174	C₂²	#90489	1	4	yes	—
175	C₂²	#90503	1	8	yes	—
176	C₂²	C₂×#6548	1	2	yes	—
177	C₂²	C₂×#6551	1	4	yes	—
178	C₂²	C₂⁴×Dic₇	1	1	yes	characteristic
179	C₄	—	2	4	no	—
180	C₄	—	2	16	no	—
181	C₄	—	2	8	no	—
182	C₄	—	28	16	no	—
183	C₂	#11740676	1	8	yes	direct product factor
184	C₂	C₂×#90426	1	4	yes	—
185	C₂	C₂×#90489	1	1	yes	characteristic
186	C₂	C₂×#90503	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₇
commutator	C₂²×C₇	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₇
Fitting	C₂²×C₇×C₄⋊Q₈	C₂
socle, prime radical	C₂⁴×C₇	C₂⁴
cosocle	C₂³×C₇	C₂⁵
hypercenter	C₂²×C₄⋊Q₈	D₇
hypocenter	C₇	C₂×#2381
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. #11740676	C₂	direct (N × H)
7. C₂×C₇×C₄⋊Q₈	C₄	semidirect
28. C₇×C₄⋊Q₈	C₂×C₄	semidirect
68. C₂×C₄⋊Q₈	Dic₇	semidirect
106. C₄⋊Q₈	C₂×Dic₇	semidirect
150. C₂×C₇	#2381	semidirect
172. C₇	C₂×#2381	semidirect
183. C₂	#11740676	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×C₇×C₄⋊Q₈	2
C₂×#90002	2
#11740676	2
C₂×#89753	2
C₂×#89709	2
C₂×#2381	7

Subgroups — C2×#11740676 #13516431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11740676 #13516431