Subgroups — C₂×#11726832 #13507100

Order 1792.

798 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₅₆	C₂	1	2	yes	maximal, complemented
3	C₂×C₇×#346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#89304	C₂	1	2	yes	maximal, complemented
5	C₂×#89300	C₂	1	1	yes	maximal, characteristic, complemented
6	#11726832	C₂	1	8	yes	maximal, direct product factor
7	C₂²×D₅₆	C₂²	1	2	yes	—
8	C₂×C₄×D₂₈	—	2	2	no	—
9	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6395	—	2	2	no	—
13	C₂×#6392	C₂²	1	2	yes	—
14	C₂×D₁₁₂	C₄	1	8	yes	complemented
15	C₂×D₁₁₂	C₂²	1	4	yes	—
16	C₄×D₅₆	—	2	4	no	—
17	C₄×D₅₆	C₂²	1	8	yes	complemented
18	C₇×#346	C₂²	1	4	yes	complemented
19	#89304	C₂²	1	8	yes	complemented
20	#89300	C₂²	1	4	yes	complemented
21	C₂×#2177	—	7	1	no	maximal, Sylow
22	C₂²×D₂₈	—	2	2	no	—
23	C₂²×C₄×D₇	—	4	2	no	—
24	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
25	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₇	D₄	1	1	yes	characteristic
27	C₂×(C₂×D₇)⋊C₄	—	4	2	no	—
28	C₂×C₄⋊Dic₇	—	2	2	no	—
29	C₂×D₅₆	—	2	2	no	—
30	C₂×D₅₆	—	2	4	no	—
31	C₂×D₅₆	C₂×C₄	1	8	yes	complemented
32	C₂×D₅₆	C₂³	1	4	yes	—
33	C₄×D₂₈	—	2	8	no	—
34	C₄×D₂₈	—	4	4	no	—
35	C₂×C₇×C₁₆	—	2	2	no	—
36	C₂×C₇×C₁₆	C₂×C₄	1	4	yes	complemented
37	C₂×C₇×C₁₆	C₂³	1	2	yes	—
38	C₂×C₇×C₁₆	C₂³	1	2	yes	—
39	C₄×C₇×C₈	D₄	1	2	yes	—
40	C₄×C₇×C₈	C₂³	1	2	yes	—
41	#6395	—	2	8	no	—
42	#6392	—	2	2	no	—
43	#6392	C₂³	1	4	yes	—
44	D₁₁₂	C₂×C₄	1	16	yes	complemented
45	C₂²×D₁₆	—	7	1	no	—
46	C₂×C₄×D₈	—	7	2	no	—
47	#2177	—	7	8	no	—
48	C₂×#367	—	7	1	no	—
49	C₂×#357	—	7	2	no	—
50	C₂×#346	D₇	1	1	yes	characteristic, complemented
51	C₂³×D₇	—	4	2	no	—
52	C₂²×C₄×C₇	D₈	1	2	yes	—
53	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×Dic₇	—	4	2	no	—
55	C₂×D₂₈	—	2	8	no	—
56	C₂×D₂₈	—	2	4	no	—
57	C₂×D₂₈	—	4	2	no	—
58	C₂×D₂₈	—	4	4	no	—
59	C₂×C₄×D₇	—	4	8	no	—
60	C₂×C₄×D₇	—	8	8	no	—
61	C₂×C₇×C₈	—	2	2	no	—
62	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
63	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
64	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
65	C₂×C₇×C₈	#41	1	2	yes	—
66	C₂×C₇×C₈	C₂⁴	1	1	yes	characteristic
67	C₄²×C₇	D₈	1	2	yes	—
68	C₄²×C₇	C₂×D₄	1	2	yes	—
69	(C₂×D₇)⋊C₄	—	4	8	no	—
70	C₄⋊Dic₇	—	2	4	no	—
71	C₄⋊Dic₇	—	4	2	no	—
72	D₅₆	—	2	8	no	—
73	D₅₆	—	2	4	no	—
74	D₅₆	C₂²×C₄	1	8	yes	complemented
75	C₇×C₁₆	—	2	2	no	—
76	C₇×C₁₆	C₂²×C₄	1	4	yes	complemented
77	C₂²×D₈	—	7	2	no	—
78	C₂×C₄×D₄	—	14	2	no	—
79	C₂×D₁₆	—	7	8	no	—
80	C₂×D₁₆	—	7	4	no	—
81	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
82	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
83	C₄×D₈	—	7	8	no	—
84	C₄×D₈	—	14	4	no	—
85	C₂×#107	—	7	2	no	—
86	C₂×#102	—	14	2	no	—
87	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
88	#367	—	7	4	no	—
89	#357	—	7	8	no	—
90	#346	C₂×D₇	1	4	yes	complemented
91	C₂³×C₇	C₂×D₈	1	1	yes	characteristic
92	C₂²×D₇	—	4	4	no	—
93	C₂²×D₇	—	4	8	no	—
94	C₂²×D₇	—	8	4	no	—
95	C₂²×D₇	—	8	4	no	—
96	C₂×C₄×C₇	—	2	4	no	—
97	C₂×C₄×C₇	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₇	C₂×D₈	1	4	yes	—
99	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
100	C₂×C₄×C₇	#135	1	2	yes	—
101	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₇	—	4	4	no	—
103	C₂×Dic₇	—	8	4	no	—
104	D₂₈	—	2	8	no	—
105	D₂₈	—	4	4	no	—
106	D₂₈	—	4	8	no	—
107	C₄×D₇	—	8	16	no	—
108	C₇×C₈	—	2	2	no	—
109	C₇×C₈	C₄×D₄	1	2	yes	—
110	C₇×C₈	C₂³×C₄	1	1	yes	characteristic
111	C₇×C₈	C₂×#41	1	1	yes	characteristic
112	C₂²×D₄	—	14	2	no	—
113	C₂³×C₄	—	28	2	no	—
114	C₂×D₈	—	7	8	no	—
115	C₂×D₈	—	7	4	no	—
116	C₂×D₈	—	14	2	no	—
117	C₂×D₈	—	14	4	no	—
118	C₂²×C₈	D₂₈	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
120	C₄×D₄	—	14	8	no	—
121	C₄×D₄	—	28	4	no	—
122	C₂×C₄⋊C₄	—	14	2	no	—
123	C₂×C₂²⋊C₄	—	28	2	no	—
124	C₂×C₄²	D₂₈	1	1	yes	characteristic
125	D₁₆	—	7	16	no	—
126	C₂×C₁₆	—	2	2	no	—
127	C₂×C₁₆	C₄×D₇	1	4	yes	complemented
128	C₂×C₁₆	C₂²×D₇	1	2	yes	—
129	C₂×C₁₆	C₂²×D₇	1	2	yes	—
130	#107	—	7	4	no	—
131	#107	—	14	2	no	—
132	#102	—	14	8	no	—
133	C₄×C₈	D₂₈	1	2	yes	—
134	C₄×C₈	C₂²×D₇	1	2	yes	—
135	C₂²×C₇	C₄×D₈	1	2	yes	—
136	C₂²×C₇	#509	1	4	yes	—
137	C₂²×C₇	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₇	—	4	8	no	—
139	C₂×D₇	—	8	4	no	—
140	C₂×D₇	—	8	4	no	—
141	C₂×D₇	—	8	16	no	—
142	C₄×C₇	—	2	4	no	—
143	C₄×C₇	C₄×D₈	1	2	yes	—
144	C₄×C₇	C₂×C₄×D₄	1	1	yes	characteristic
145	C₄×C₇	C₂×#135	1	1	yes	characteristic
146	Dic₇	—	8	4	no	—
147	C₂⁴	—	28	2	no	—
148	C₂×D₄	—	14	8	no	—
149	C₂×D₄	—	14	4	no	—
150	C₂×D₄	—	28	2	no	—
151	C₂×D₄	—	28	4	no	—
152	C₂²×C₄	—	28	2	no	—
153	C₂²×C₄	—	28	8	no	—
154	C₂²×C₄	—	56	8	no	—
155	C₂²×C₄	D₅₆	1	2	yes	—
156	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
157	D₈	—	7	8	no	—
158	D₈	—	14	8	no	—
159	D₈	—	14	4	no	—
160	C₂×C₈	—	2	2	no	—
161	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
162	C₂×C₈	C₂×D₂₈	1	2	yes	—
163	C₂×C₈	C₂×D₂₈	1	1	yes	characteristic
164	C₂×C₈	#1157	1	2	yes	—
165	C₂×C₈	C₂³×D₇	1	1	yes	characteristic
166	C₄⋊C₄	—	14	4	no	—
167	C₄⋊C₄	—	28	2	no	—
168	C₂²⋊C₄	—	28	8	no	—
169	C₄²	D₅₆	1	2	yes	—
170	C₄²	C₂×D₂₈	1	2	yes	—
171	C₁₆	—	2	2	no	—
172	C₁₆	C₂×C₄×D₇	1	4	yes	complemented
173	C₂×C₇	#2177	1	4	yes	complemented
174	C₂×C₇	C₂×C₄×D₈	1	1	yes	characteristic
175	C₂×C₇	C₂×#509	1	2	yes	—
176	D₇	—	8	8	no	—
177	C₂³	—	28	4	no	—
178	C₂³	—	28	8	no	—
179	C₂³	—	56	4	no	—
180	C₂³	—	56	4	no	—
181	C₂³	C₂×D₅₆	1	1	yes	characteristic
182	D₄	—	14	8	no	—
183	D₄	—	28	4	no	—
184	D₄	—	28	8	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	28	4	no	—
187	C₂×C₄	—	56	16	no	—
188	C₂×C₄	—	56	4	no	—
189	C₂×C₄	C₄×D₂₈	1	2	yes	—
190	C₂×C₄	C₂×D₅₆	1	4	yes	—
191	C₂×C₄	C₂×D₅₆	1	1	yes	characteristic
192	C₂×C₄	#6467	1	2	yes	—
193	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	C₄×D₂₈	1	2	yes	—
196	C₈	C₂²×C₄×D₇	1	1	yes	characteristic
197	C₈	C₂×#1157	1	1	yes	characteristic
198	C₇	C₂×#2177	1	1	yes	characteristic, complemented, Sylow
199	C₂²	—	28	8	no	—
200	C₂²	—	56	4	no	—
201	C₂²	—	56	4	no	—
202	C₂²	—	56	16	no	—
203	C₂²	C₄×D₅₆	1	2	yes	—
204	C₂²	#89680	1	4	yes	—
205	C₂²	C₂²×D₅₆	1	1	yes	characteristic
206	C₄	—	2	4	no	—
207	C₄	—	56	4	no	—
208	C₄	C₄×D₅₆	1	2	yes	—
209	C₄	C₂×C₄×D₂₈	1	1	yes	characteristic
210	C₄	C₂×#6467	1	1	yes	characteristic
211	C₂	—	56	8	no	—
212	C₂	#11726832	1	4	yes	direct product factor
213	C₂	C₂×C₄×D₅₆	1	1	yes	characteristic
214	C₂	C₂×#89680	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₅₆
commutator	C₇×C₈	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇
Fitting	C₂×C₇×#346	C₂
socle	C₂³×C₇	C₂×D₈
prime radical	C₂²×D₁₁₂	C₂
cosocle	C₂×C₇×C₈	C₂⁴
hypercenter	C₂×#346	D₇
hypocenter	C₇	C₂×#2177
LCS	G ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₅₆	C₂	semidirect
3. C₂×C₇×#346	C₂	semidirect
4. C₂×#89304	C₂	semidirect
5. C₂×#89300	C₂	semidirect
6. #11726832	C₂	direct (N × H)
14. C₂×D₁₁₂	C₄	semidirect
17. C₄×D₅₆	C₂²	semidirect
18. C₇×#346	C₂²	semidirect
19. #89304	C₂²	semidirect
20. #89300	C₂²	semidirect
31. C₂×D₅₆	C₂×C₄	semidirect
36. C₂×C₇×C₁₆	C₂×C₄	semidirect
44. D₁₁₂	C₂×C₄	semidirect
50. C₂×#346	D₇	semidirect
74. D₅₆	C₂²×C₄	semidirect
76. C₇×C₁₆	C₂²×C₄	semidirect
90. #346	C₂×D₇	semidirect
127. C₂×C₁₆	C₄×D₇	semidirect
172. C₁₆	C₂×C₄×D₇	semidirect
173. C₂×C₇	#2177	semidirect
198. C₇	C₂×#2177	semidirect
212. C₂	#11726832	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×D₁₁₂	2
C₂×C₄×D₅₆	2
C₂×C₇×#346	2
C₂×#89304	2
C₂×#89300	2
#11726832	2
C₂×#2177	7

Subgroups — C2×#11726832 #13507100

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11726832 #13507100