Subgroups — C₂×#11729108 #13509275

Order 1792.

494 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	C₂×C₇×#499	C₂	1	1	yes	maximal, characteristic
2	C₂×#89455	C₂	1	1	yes	maximal, characteristic
3	#11729108	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89285	C₂	1	2	yes	maximal
5	C₂×#89284	C₂	1	2	yes	maximal
6	C₂×C₄×C₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂×C₇×#125	—	2	1	no	—
8	C₂×#6433	—	2	1	no	—
9	C₇×#499	—	2	2	no	—
10	C₇×#499	C₂²	1	4	yes	—
11	C₂×C₇×#107	C₂²	1	2	yes	—
12	C₂×C₇×#106	—	2	1	no	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6392	C₂²	1	2	yes	—
15	C₂×#6391	—	2	1	no	—
16	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
17	C₂²×C₇⋊C₁₆	C₂²	1	1	yes	characteristic
18	#89455	—	2	2	no	—
19	#89455	C₂²	1	4	yes	—
20	#89285	C₂²	1	8	yes	—
21	#89284	C₂²	1	8	yes	—
22	C₄×C₇⋊C₁₆	C₂²	1	4	yes	—
23	C₂×#2253	—	7	1	no	maximal, Sylow
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₇×#125	—	2	4	no	—
27	C₇×#125	—	4	2	no	—
28	C₂×C₇×C₄⋊C₄	—	2	2	no	—
29	C₂×C₇×C₄⋊C₄	—	4	2	no	—
30	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
31	C₂×C₄⋊Dic₇	—	2	2	no	—
32	C₂×Dic₇⋊C₄	—	4	2	no	—
33	#6433	—	2	4	no	—
34	#6433	—	4	2	no	—
35	C₇×#107	—	2	2	no	—
36	C₇×#107	C₂³	1	4	yes	—
37	C₇×#106	—	2	4	no	—
38	C₄×C₇×C₈	D₄	1	2	yes	—
39	C₄×C₇×C₈	C₂³	1	2	yes	—
40	#6392	—	2	2	no	—
41	#6392	C₂³	1	4	yes	—
42	#6391	—	2	4	no	—
43	C₂×C₇⋊C₁₆	Q₈	1	4	yes	—
44	C₂×C₇⋊C₁₆	Q₈	1	4	yes	—
45	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
46	C₂×C₇⋊C₁₆	C₂³	1	2	yes	—
47	C₂×#499	—	7	1	no	—
48	C₂×#499	D₇	1	1	yes	characteristic
49	#2253	—	7	8	no	—
50	C₂×#367	—	7	2	no	—
51	C₂×#366	—	7	2	no	—
52	C₂×C₄×C₁₆	—	7	1	no	—
53	C₂²×C₄×C₇	—	4	2	no	—
54	C₂²×C₄×C₇	D₈	1	2	yes	—
55	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
56	C₂²×Dic₇	—	4	2	no	—
57	C₂×C₇×C₈	Q₁₆	1	4	yes	—
58	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
59	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
60	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
61	C₂×C₇×C₈	C₂×Q₈	1	2	yes	—
62	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
63	C₇×C₄⋊C₄	—	2	4	no	—
64	C₇×C₄⋊C₄	—	4	2	no	—
65	C₇×C₄⋊C₄	—	4	8	no	—
66	C₄²×C₇	D₈	1	2	yes	—
67	C₄²×C₇	C₂×D₄	1	2	yes	—
68	C₄⋊Dic₇	—	2	4	no	—
69	C₄⋊Dic₇	—	4	2	no	—
70	Dic₇⋊C₄	—	4	8	no	—
71	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
72	C₇⋊C₁₆	C₂×Q₈	1	4	yes	—
73	C₂×#125	—	2	1	no	—
74	C₂×#125	—	14	1	no	—
75	C₂²×C₁₆	—	7	1	no	—
76	C₂²×C₁₆	—	7	1	no	—
77	#499	—	2	2	no	—
78	#499	—	7	4	no	—
79	#499	—	14	2	no	—
80	#499	C₂×D₇	1	4	yes	—
81	C₂×#107	—	7	2	no	—
82	C₂×#107	C₂×D₇	1	2	yes	—
83	C₂×#106	—	2	1	no	—
84	C₂×#106	—	14	1	no	—
85	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
86	#367	—	7	8	no	—
87	#366	—	7	8	no	—
88	C₄×C₁₆	—	7	4	no	—
89	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₇	—	2	4	no	—
91	C₂×C₄×C₇	—	4	4	no	—
92	C₂×C₄×C₇	—	8	4	no	—
93	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
95	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
96	C₂×C₄×C₇	C₂×Q₁₆	1	2	yes	—
97	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
98	C₂×Dic₇	—	4	4	no	—
99	C₂×Dic₇	—	8	4	no	—
100	C₇×C₈	C₄⋊Q₈	1	2	yes	—
101	C₇×C₈	C₂×Q₁₆	1	4	yes	—
102	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
103	C₇×C₈	C₂²×Q₈	1	1	yes	characteristic
104	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
105	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
106	#125	—	2	4	no	—
107	#125	—	4	2	no	—
108	#125	—	14	4	no	—
109	#125	—	28	2	no	—
110	C₂×C₄⋊C₄	—	2	2	no	—
111	C₂×C₄⋊C₄	—	4	2	no	—
112	C₂×C₄⋊C₄	—	14	2	no	—
113	C₂×C₄⋊C₄	—	28	2	no	—
114	C₂×C₄²	C₂²⋊D₇	1	1	yes	characteristic
115	C₂×C₁₆	—	7	4	no	—
116	C₂×C₁₆	—	7	4	no	—
117	C₂×C₁₆	—	7	2	no	—
118	C₂×C₁₆	—	7	2	no	—
119	#107	—	2	2	no	—
120	#107	—	7	4	no	—
121	#107	—	14	2	no	—
122	#107	C₂²×D₇	1	4	yes	—
123	#106	—	2	4	no	—
124	#106	—	14	4	no	—
125	C₄×C₈	C₂²⋊D₇	1	2	yes	—
126	C₄×C₈	C₂²×D₇	1	2	yes	—
127	C₂²×C₇	#500	1	2	yes	—
128	C₂²×C₇	#508	1	4	yes	—
129	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
130	C₄×C₇	—	2	4	no	—
131	C₄×C₇	—	8	4	no	—
132	C₄×C₇	#500	1	2	yes	—
133	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
134	C₄×C₇	C₂²×Q₁₆	1	1	yes	characteristic
135	Dic₇	—	8	4	no	—
136	C₂²×C₄	—	4	2	no	—
137	C₂²×C₄	—	28	2	no	—
138	C₂²×C₄	#1141	1	2	yes	—
139	C₂²×C₄	C₂×C₂²⋊D₇	1	1	yes	characteristic
140	C₂×C₈	#1144	1	4	yes	—
141	C₂×C₈	Q₈×D₇	1	2	yes	—
142	C₂×C₈	Q₈×D₇	1	2	yes	—
143	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
144	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
145	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	4	2	no	—
148	C₄⋊C₄	—	4	8	no	—
149	C₄⋊C₄	—	14	4	no	—
150	C₄⋊C₄	—	28	8	no	—
151	C₄⋊C₄	—	28	2	no	—
152	C₄²	#1141	1	2	yes	—
153	C₄²	C₂×C₂²⋊D₇	1	2	yes	—
154	C₁₆	—	7	4	no	—
155	C₁₆	—	7	4	no	—
156	C₂×C₇	#2253	1	4	yes	complemented
157	C₂×C₇	C₂×#500	1	1	yes	characteristic
158	C₂×C₇	C₂×#508	1	2	yes	—
159	C₂³	C₂×#1141	1	1	yes	characteristic
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	4	4	no	—
162	C₂×C₄	—	8	4	no	—
163	C₂×C₄	—	28	4	no	—
164	C₂×C₄	—	56	4	no	—
165	C₂×C₄	C₂×#1141	1	4	yes	—
166	C₂×C₄	C₂×#1141	1	1	yes	characteristic
167	C₂×C₄	C₂×#1144	1	2	yes	—
168	C₂×C₄	#6507	1	2	yes	—
169	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
170	C₈	C₂×#1144	1	4	yes	—
171	C₈	#6507	1	2	yes	—
172	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
173	C₈	C₂×Q₈×D₇	1	1	yes	characteristic
174	C₇	C₂×#2253	1	1	yes	characteristic, complemented, Sylow
175	C₂²	#89850	1	2	yes	—
176	C₂²	#89966	1	4	yes	—
177	C₂²	C₂²×#1141	1	1	yes	characteristic
178	C₄	—	2	4	no	—
179	C₄	—	8	4	no	—
180	C₄	—	56	4	no	—
181	C₄	#89850	1	2	yes	—
182	C₄	C₂²×#1144	1	1	yes	characteristic
183	C₄	C₂×#6507	1	1	yes	characteristic
184	C₂	#11729108	1	4	yes	direct product factor
185	C₂	C₂×#89850	1	1	yes	characteristic
186	C₂	C₂×#89966	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#1141
commutator, cosocle	C₂×C₇×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#499	C₂
socle, prime radical	C₂³×C₇	C₂×D₈
hypercenter	C₂×#499	D₇
hypocenter	C₇	C₂×#2253
LCS	G ▷ C₂×C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#499

Complements

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

N	H ≅ G/N	type
3. #11729108	C₂	direct (N × H)
156. C₂×C₇	#2253	semidirect
174. C₇	C₂×#2253	semidirect
184. C₂	#11729108	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₇×#499	2
C₂×#89455	2
#11729108	2
C₂×#89285	2
C₂×#89284	2
C₂×C₄×C₇⋊C₁₆	2
C₂×#2253	7

Subgroups — C2×#11729108 #13509275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11729108 #13509275