Subgroups — C₂×#11727596 #13354256

Order 1792.

1006 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₇×#97	C₂	1	1	yes	maximal, characteristic
2	C₂²×#6393	C₂	1	4	yes	maximal
3	C₂×#89696	C₂	1	2	yes	maximal
4	#11727596	C₂	1	8	yes	maximal, direct product factor
5	C₂³×C₇×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
7	C₂²×#1136	C₂²	1	4	yes	—
8	C₂×C₇×#97	C₄	1	4	yes	—
9	C₂×C₇×#97	C₄	1	4	yes	—
10	C₂×C₇×#97	C₂²	1	4	yes	—
11	C₂×#6393	—	2	16	no	—
12	C₂×#6393	C₂²	1	16	yes	—
13	#89696	—	2	4	no	—
14	#89696	C₂²	1	8	yes	—
15	C₂×C₄⋊(C₇⋊C₈)	—	2	4	no	—
16	C₂×#1849	—	7	1	no	maximal, Sylow
17	C₂³×C₄×C₇	Q₈	1	2	yes	—
18	C₂³×C₄×C₇	C₂³	1	1	yes	characteristic
19	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
20	C₂²×C₇×C₈	C₂×C₄	1	4	yes	—
21	C₂²×C₇×C₈	D₄	1	8	yes	—
22	C₂²×C₇×C₈	Q₈	1	8	yes	—
23	C₂²×C₇×C₈	C₂³	1	4	yes	—
24	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₇	D₄	1	2	yes	—
27	C₂×C₄²×C₇	D₄	1	2	yes	—
28	C₂×C₄²×C₇	Q₈	1	2	yes	—
29	C₂×C₄²×C₇	C₂³	1	2	yes	—
30	C₂×#1136	—	2	4	no	—
31	C₂×#1136	—	2	16	no	—
32	C₂×#1136	C₂³	1	8	yes	—
33	C₂²×C₇⋊C₈	—	2	4	no	—
34	C₇×#97	C₂×C₄	1	16	yes	—
35	#6393	—	2	32	no	—
36	C₄⋊(C₇⋊C₈)	—	2	16	no	—
37	C₂²×#108	—	7	4	no	—
38	C₂×#423	—	7	2	no	—
39	C₂²×#97	D₇	1	1	yes	characteristic
40	#1849	—	7	8	no	—
41	C₂⁴×C₇	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₇	—	4	4	no	—
43	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
44	C₂²×C₄×C₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
47	C₂²×C₄×C₇	C₂×D₄	1	4	yes	—
48	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
51	C₂²×C₄×C₇	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₇	C₂×Q₈	1	4	yes	—
53	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
54	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
55	C₂×C₇×C₈	C₄⋊C₄	1	16	yes	—
56	C₂×C₇×C₈	C₂²×C₄	1	8	yes	—
57	C₂×C₇×C₈	C₂×D₄	1	8	yes	—
58	C₂×C₇×C₈	C₂×Q₈	1	8	yes	—
59	C₄²×C₇	C₄⋊C₄	1	4	yes	—
60	C₄²×C₇	C₄⋊C₄	1	4	yes	—
61	C₄²×C₇	C₂²×C₄	1	4	yes	—
62	C₄²×C₇	C₂×D₄	1	4	yes	—
63	#1136	—	2	16	no	—
64	#1136	—	4	8	no	—
65	C₂×C₇⋊C₈	—	2	8	no	—
66	C₂×C₇⋊C₈	—	4	8	no	—
67	C₂²×M₁₆	—	7	4	no	—
68	C₂³×C₈	C₂×D₇	1	2	yes	—
69	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
70	C₂×#108	—	7	16	no	—
71	C₂×#108	—	14	16	no	—
72	#423	—	7	8	no	—
73	#423	—	14	4	no	—
74	C₂×C₄⋊C₈	—	14	4	no	—
75	C₂×#97	Dic₇	1	4	yes	—
76	C₂×#97	Dic₇	1	4	yes	—
77	C₂×#97	C₂×D₇	1	4	yes	—
78	C₂³×C₇	—	2	4	no	—
79	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂³×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
81	C₂³×C₇	C₄⋊Q₈	1	4	yes	—
82	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
83	C₂×C₄×C₇	—	2	8	no	—
84	C₂×C₄×C₇	—	4	4	no	—
85	C₂×C₄×C₇	—	4	4	no	—
86	C₂×C₄×C₇	C₂×C₄⋊C₄	1	8	yes	—
87	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
88	C₂×C₄×C₇	C₂×C₄⋊C₄	1	4	yes	—
89	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₇	C₂×C₄⋊C₄	1	1	yes	characteristic
91	C₂×C₄×C₇	#127	1	2	yes	—
92	C₂×C₄×C₇	#127	1	2	yes	—
93	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
94	C₂×C₄×C₇	C₄⋊Q₈	1	2	yes	—
95	C₂×C₄×C₇	C₄⋊Q₈	1	4	yes	—
96	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
97	C₂×C₄×C₇	C₂²×D₄	1	2	yes	—
98	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
99	C₂×C₄×C₇	C₂²×Q₈	1	2	yes	—
100	C₇×C₈	C₂×C₄⋊C₄	1	16	yes	—
101	C₇⋊C₈	—	4	8	no	—
102	C₂³×C₄	Dic₁₄	1	2	yes	—
103	C₂³×C₄	C₂²×D₇	1	1	yes	characteristic
104	C₂×M₁₆	—	7	8	no	—
105	C₂×M₁₆	—	14	4	no	—
106	C₂×M₁₆	—	14	16	no	—
107	C₂²×C₈	—	14	4	no	—
108	C₂²×C₈	Dic₁₄	1	8	yes	—
109	C₂²×C₈	D₂₈	1	8	yes	—
110	C₂²×C₈	C₂×Dic₇	1	4	yes	—
111	C₂²×C₈	C₂×Dic₇	1	4	yes	—
112	C₂²×C₈	C₂²×D₇	1	4	yes	—
113	C₂×C₄²	Dic₁₄	1	2	yes	—
114	C₂×C₄²	D₂₈	1	2	yes	—
115	C₂×C₄²	D₂₈	1	2	yes	—
116	C₂×C₄²	C₂×Dic₇	1	2	yes	—
117	C₂×C₄²	C₂×Dic₇	1	2	yes	—
118	C₂×C₄²	C₂²×D₇	1	2	yes	—
119	#108	—	14	32	no	—
120	C₄⋊C₈	—	14	16	no	—
121	#97	C₂×Dic₇	1	16	yes	—
122	C₂²×C₇	—	2	8	no	—
123	C₂²×C₇	—	2	2	no	—
124	C₂²×C₇	—	2	2	no	—
125	C₂²×C₇	#384	1	2	yes	—
126	C₂²×C₇	#384	1	2	yes	—
127	C₂²×C₇	#430	1	4	yes	—
128	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂²×C₇	C₂×C₄⋊Q₈	1	2	yes	—
130	C₄×C₇	—	2	8	no	—
131	C₄×C₇	#384	1	2	yes	—
132	C₄×C₇	#384	1	2	yes	—
133	C₄×C₇	C₂²×C₄⋊C₄	1	2	yes	—
134	C₄×C₇	C₂×#127	1	1	yes	characteristic
135	C₄×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂⁴	C₂×Dic₁₄	1	1	yes	characteristic
137	C₂²×C₄	—	4	4	no	—
138	C₂²×C₄	C₄⋊Dic₇	1	4	yes	—
139	C₂²×C₄	C₄⋊Dic₇	1	4	yes	—
140	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
141	C₂²×C₄	C₂×Dic₁₄	1	1	yes	characteristic
142	C₂²×C₄	C₂×Dic₁₄	1	4	yes	—
143	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
144	C₂²×C₄	C₂×D₂₈	1	4	yes	—
145	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
146	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
147	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
148	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
149	M₁₆	—	14	16	no	—
150	M₁₆	—	28	8	no	—
151	C₂×C₈	—	14	8	no	—
152	C₂×C₈	—	28	8	no	—
153	C₂×C₈	C₄⋊Dic₇	1	16	yes	—
154	C₂×C₈	C₄⋊Dic₇	1	16	yes	—
155	C₂×C₈	C₂×Dic₁₄	1	8	yes	—
156	C₂×C₈	C₂×D₂₈	1	8	yes	—
157	C₂×C₈	C₂²×Dic₇	1	8	yes	—
158	C₄²	C₄⋊Dic₇	1	4	yes	—
159	C₄²	C₄⋊Dic₇	1	4	yes	—
160	C₄²	C₂×D₂₈	1	4	yes	—
161	C₄²	C₂²×Dic₇	1	4	yes	—
162	C₂×C₇	—	2	2	no	—
163	C₂×C₇	—	2	2	no	—
164	C₂×C₇	#1849	1	4	yes	complemented
165	C₂×C₇	C₂×#384	1	1	yes	characteristic
166	C₂×C₇	C₂×#430	1	2	yes	—
167	C₂³	—	2	4	no	—
168	C₂³	#6432	1	4	yes	—
169	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
170	C₂³	C₂×C₄⋊Dic₇	1	1	yes	characteristic
171	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
172	C₂×C₄	—	2	8	no	—
173	C₂×C₄	—	4	4	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	#6432	1	2	yes	—
176	C₂×C₄	#6432	1	2	yes	—
177	C₂×C₄	#6432	1	4	yes	—
178	C₂×C₄	#6437	1	2	yes	—
179	C₂×C₄	#6437	1	2	yes	—
180	C₂×C₄	C₂×C₄⋊Dic₇	1	8	yes	—
181	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
182	C₂×C₄	C₂×C₄⋊Dic₇	1	1	yes	characteristic
183	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
184	C₂×C₄	C₂×C₄⋊Dic₇	1	4	yes	—
185	C₂×C₄	C₂²×Dic₁₄	1	2	yes	—
186	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
187	C₂×C₄	C₂²×D₂₈	1	2	yes	—
188	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
189	C₈	—	28	8	no	—
190	C₈	C₂×C₄⋊Dic₇	1	16	yes	—
191	C₇	C₂×#1849	1	1	yes	characteristic, complemented, Sylow
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	8	no	—
195	C₂²	#89707	1	2	yes	—
196	C₂²	#89707	1	2	yes	—
197	C₂²	#89897	1	4	yes	—
198	C₂²	C₂×#6432	1	2	yes	—
199	C₂²	C₂²×C₄⋊Dic₇	1	1	yes	characteristic
200	C₄	—	2	8	no	—
201	C₄	#89707	1	2	yes	—
202	C₄	#89707	1	2	yes	—
203	C₄	C₂×#6432	1	1	yes	characteristic
204	C₄	C₂×#6437	1	1	yes	characteristic
205	C₄	C₂²×C₄⋊Dic₇	1	2	yes	—
206	C₂	—	2	2	no	—
207	C₂	—	2	2	no	—
208	C₂	#11727596	1	4	yes	direct product factor
209	C₂	C₂×#89707	1	1	yes	characteristic
210	C₂	C₂×#89897	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₂²×C₇×#97	C₂
socle	C₂³×C₇	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇	C₂×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	C₂²×#97	D₇
hypocenter	C₇	C₂×#1849
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
4. #11727596	C₂	direct (N × H)
164. C₂×C₇	#1849	semidirect
191. C₇	C₂×#1849	semidirect
208. C₂	#11727596	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×#97	2
C₂²×#6393	2
C₂×#89696	2
#11727596	2
C₂×#1849	7

Subgroups — C2×#11727596 #13354256

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727596 #13354256