Subgroups — C₂×C₇⋊#338 #11729751

Order 896.

372 conjugacy classes of subgroups.

The subgroup lattice has 178 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×#1136	C₂	1	2	yes	maximal
3	C₇⋊#338	C₂	1	4	yes	maximal, direct product factor
4	C₂³×C₄×C₇	C₂²	1	1	yes	characteristic
5	C₂×C₇×M₁₆	—	2	2	no	—
6	C₂×C₇×M₁₆	—	2	2	no	—
7	C₂×C₇×M₁₆	C₄	1	2	yes	—
8	C₂×C₇×M₁₆	C₂²	1	2	yes	—
9	C₂²×C₇×C₈	C₄	1	2	yes	—
10	C₂×#1136	—	2	2	no	—
11	C₂×#1136	—	2	2	no	—
12	C₂×#1136	—	2	2	no	—
13	C₂×#1136	—	2	2	no	—
14	C₂×#1136	C₄	1	4	yes	—
15	C₂×#1136	C₂²	1	4	yes	—
16	C₂²×C₇⋊C₈	C₄	1	4	yes	—
17	C₂×#338	—	7	1	no	maximal, Sylow
18	C₂⁴×C₇	Q₈	1	1	yes	characteristic
19	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₇	D₄	1	2	yes	—
26	C₂²×C₄×C₇	D₄	1	2	yes	—
27	C₂²×C₄×C₇	Q₈	1	1	yes	characteristic
28	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
29	C₇×M₁₆	—	2	4	no	—
30	C₇×M₁₆	—	2	4	no	—
31	C₂×C₇×C₈	—	2	2	no	—
32	C₂×C₇×C₈	—	2	2	no	—
33	C₂×C₇×C₈	C₂×C₄	1	4	yes	—
34	#1136	—	2	4	no	—
35	#1136	—	2	4	no	—
36	#1136	—	2	4	no	—
37	#1136	—	2	4	no	—
38	C₂×C₇⋊C₈	—	2	4	no	—
39	C₂×C₇⋊C₈	—	2	4	no	—
40	C₂×C₇⋊C₈	C₂×C₄	1	8	yes	—
41	C₂²×M₁₆	—	7	2	no	—
42	C₂²×M₁₆	D₇	1	1	yes	characteristic
43	#338	—	7	4	no	—
44	C₂³×C₇	—	4	2	no	—
45	C₂³×C₇	C₄⋊C₄	1	2	yes	—
46	C₂³×C₇	C₄⋊C₄	1	2	yes	—
47	C₂³×C₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂³×C₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂³×C₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₇	C₄²	1	2	yes	—
51	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
52	C₂×C₄×C₇	C₄²	1	1	yes	characteristic
53	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
54	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
57	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₇	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₇	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
61	C₂×C₄×C₇	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₇	C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₄×C₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
68	C₇×C₈	—	2	4	no	—
69	C₇⋊C₈	—	2	8	no	—
70	C₂³×C₄	C₂×D₇	1	1	yes	characteristic
71	C₂×M₁₆	—	2	2	no	—
72	C₂×M₁₆	—	2	2	no	—
73	C₂×M₁₆	—	7	4	no	—
74	C₂×M₁₆	—	7	4	no	—
75	C₂×M₁₆	—	14	2	no	—
76	C₂×M₁₆	—	14	2	no	—
77	C₂×M₁₆	—	14	2	no	—
78	C₂×M₁₆	—	14	2	no	—
79	C₂×M₁₆	Dic₇	1	2	yes	—
80	C₂×M₁₆	C₂×D₇	1	2	yes	—
81	C₂²×C₈	—	7	4	no	—
82	C₂²×C₈	Dic₇	1	2	yes	—
83	C₂²×C₇	—	2	4	no	—
84	C₂²×C₇	—	2	2	no	—
85	C₂²×C₇	—	4	2	no	—
86	C₂²×C₇	—	4	1	no	—
87	C₂²×C₇	—	4	1	no	—
88	C₂²×C₇	#95	1	2	yes	—
89	C₂²×C₇	#95	1	1	yes	characteristic
90	C₂²×C₇	#95	1	1	yes	characteristic
91	C₂²×C₇	C₂×C₄⋊C₄	1	2	yes	—
92	C₂²×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₄×C₇	#95	1	2	yes	—
94	C₄×C₇	#95	1	1	yes	characteristic
95	C₄×C₇	#95	1	1	yes	characteristic
96	C₄×C₇	C₂×C₄²	1	1	yes	characteristic
97	C₄×C₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂⁴	Dic₁₄	1	1	yes	characteristic
100	C₂²×C₄	Dic₁₄	1	1	yes	characteristic
101	C₂²×C₄	C₄×D₇	1	2	yes	—
102	C₂²×C₄	C₄×D₇	1	2	yes	—
103	C₂²×C₄	D₂₈	1	1	yes	characteristic
104	C₂²×C₄	D₂₈	1	1	yes	characteristic
105	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
106	C₂²×C₄	C₂×Dic₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
108	C₂²×C₄	C₂²⋊D₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	2	4	no	—
112	M₁₆	—	14	4	no	—
113	M₁₆	—	14	4	no	—
114	M₁₆	—	14	4	no	—
115	M₁₆	—	14	4	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	7	8	no	—
119	C₂×C₈	—	14	4	no	—
120	C₂×C₈	—	14	4	no	—
121	C₂×C₈	C₂×Dic₇	1	4	yes	—
122	C₂×C₇	—	2	2	no	—
123	C₂×C₇	—	2	1	no	—
124	C₂×C₇	—	2	2	no	—
125	C₂×C₇	—	2	1	no	—
126	C₂×C₇	#338	1	2	yes	complemented
127	C₂×C₇	C₂×#95	1	1	yes	characteristic
128	C₂³	—	4	2	no	—
129	C₂³	Dic₇⋊C₄	1	2	yes	—
130	C₂³	Dic₇⋊C₄	1	2	yes	—
131	C₂³	C₄⋊Dic₇	1	1	yes	characteristic
132	C₂³	C₄⋊Dic₇	1	1	yes	characteristic
133	C₂³	C₂×Dic₁₄	1	1	yes	characteristic
134	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
135	C₂×C₄	C₄×Dic₇	1	2	yes	—
136	C₂×C₄	C₄×Dic₇	1	1	yes	characteristic
137	C₂×C₄	Dic₇⋊C₄	1	2	yes	—
138	C₂×C₄	Dic₇⋊C₄	1	2	yes	—
139	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
140	C₂×C₄	C₄⋊Dic₇	1	1	yes	characteristic
141	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
142	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
143	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
144	C₂×C₄	(C₂×D₇)⋊C₄	1	2	yes	—
145	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
146	C₂×C₄	C₂²⋊Dic₇	1	1	yes	characteristic
147	C₂×C₄	C₂²⋊Dic₇	1	2	yes	—
148	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
149	C₂×C₄	C₂×D₂₈	1	1	yes	characteristic
150	C₂×C₄	C₂²×Dic₇	1	1	yes	characteristic
151	C₂×C₄	C₂×C₂²⋊D₇	1	2	yes	—
152	C₈	—	2	4	no	—
153	C₈	—	14	8	no	—
154	C₇	C₂×#338	1	1	yes	characteristic, complemented, Sylow
155	C₂²	—	2	4	no	—
156	C₂²	—	2	2	no	—
157	C₂²	—	4	2	no	—
158	C₂²	—	4	1	no	—
159	C₂²	—	4	1	no	—
160	C₂²	C₇⋊#95	1	2	yes	—
161	C₂²	C₇⋊#95	1	1	yes	characteristic
162	C₂²	C₇⋊#95	1	1	yes	characteristic
163	C₂²	C₂×Dic₇⋊C₄	1	2	yes	—
164	C₂²	C₂×C₄⋊Dic₇	1	1	yes	characteristic
165	C₄	C₇⋊#95	1	2	yes	—
166	C₄	C₇⋊#95	1	1	yes	characteristic
167	C₄	C₇⋊#95	1	1	yes	characteristic
168	C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
169	C₄	C₂×(C₂×D₇)⋊C₄	1	2	yes	—
170	C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
171	C₂	—	2	2	no	—
172	C₂	—	2	1	no	—
173	C₂	—	2	2	no	—
174	C₂	—	2	1	no	—
175	C₂	C₇⋊#338	1	2	yes	direct product factor
176	C₂	C₂×C₇⋊#95	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₇
commutator	C₄×C₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₇
Fitting	C₂²×C₇×M₁₆	C₂
socle	C₂²×C₇	#95
prime radical	C₂⁴×C₇	Q₈
cosocle	C₂²×C₄×C₇	C₂³
hypercenter	C₂²×M₁₆	D₇
hypocenter	C₇	C₂×#338
LCS	G ▷ C₄×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₇⋊#338	C₂	direct (N × H)
126. C₂×C₇	#338	semidirect
154. C₇	C₂×#338	semidirect
175. C₂	C₇⋊#338	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×M₁₆	2
C₂²×#1136	2
C₇⋊#338	2
C₂×#338	7

Subgroups — C2×C7⋊#338 #11729751

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₇⋊#338 #11729751