Subgroups — C₂×#11725634 #13274255

Order 1792.

302 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×#508	C₂	1	1	yes	maximal, characteristic
2	C₂×#89300	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#89294	C₂	1	1	yes	maximal, characteristic, complemented
4	#11725634	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₇×#135	—	2	1	no	—
6	C₇×#508	—	2	1	no	—
7	C₇×#508	—	2	1	no	—
8	C₇×#508	C₄	1	2	yes	complemented
9	C₇×#508	C₂²	1	2	yes	—
10	C₂×C₇×Q₃₂	C₄	1	1	yes	characteristic, complemented
11	C₂×C₇×SD₃₂	—	2	1	no	—
12	C₂×C₇×D₁₆	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₇×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×#6392	—	2	1	no	—
15	#89300	—	2	1	no	—
16	#89300	C₂²	1	2	yes	complemented
17	#89294	—	2	1	no	—
18	#89294	C₂²	1	2	yes	complemented
19	C₂×C₇⋊C₃₂	—	2	1	no	—
20	C₂×#1418	—	7	1	no	maximal, Sylow
21	C₂×C₇×#41	—	4	1	no	—
22	C₇×#135	—	2	2	no	—
23	C₇×#135	—	2	2	no	—
24	C₇×#135	—	4	1	no	—
25	C₇×#135	—	4	1	no	—
26	C₂×C₇×Q₁₆	—	2	1	no	—
27	C₂×C₇×SD₁₆	—	4	1	no	—
28	C₂×C₇×D₈	—	2	1	no	—
29	C₂²×C₇×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄⋊Dic₇	—	4	1	no	—
31	C₇×Q₃₂	—	2	1	no	—
32	C₇×Q₃₂	C₂×C₄	1	2	yes	complemented
33	C₇×SD₃₂	—	2	2	no	—
34	C₇×SD₃₂	—	2	2	no	—
35	C₇×D₁₆	—	2	1	no	—
36	C₇×D₁₆	C₂×C₄	1	2	yes	complemented
37	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
38	C₂×C₇×C₁₆	C₂×C₄	1	1	yes	characteristic
39	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
40	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
41	C₂×C₇×C₁₆	D₄	1	1	yes	characteristic
42	C₂×C₇×C₁₆	C₂³	1	1	yes	characteristic
43	#6392	—	2	2	no	—
44	#6392	—	4	1	no	—
45	C₇⋊C₃₂	—	2	2	no	—
46	C₂×#508	D₇	1	1	yes	characteristic
47	C₂×M₆₄	—	7	1	no	—
48	C₂×#367	—	7	1	no	—
49	#1418	—	7	4	no	—
50	C₇×#41	—	4	2	no	—
51	C₇×#41	—	4	2	no	—
52	C₇×#41	—	8	1	no	—
53	C₇×#41	—	8	1	no	—
54	C₂×C₇×Q₈	—	4	1	no	—
55	C₂×C₇×D₄	—	4	1	no	—
56	C₂×C₇×D₄	—	8	1	no	—
57	C₂²×C₄×C₇	—	8	1	no	—
58	C₂²×C₄×C₇	D₈	1	1	yes	characteristic
59	C₂²×Dic₇	—	8	1	no	—
60	C₇×Q₁₆	—	2	2	no	—
61	C₇×Q₁₆	—	4	1	no	—
62	C₇×SD₁₆	—	4	2	no	—
63	C₇×SD₁₆	—	4	2	no	—
64	C₇×D₈	—	2	2	no	—
65	C₇×D₈	—	4	1	no	—
66	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₇×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₇×C₈	D₈	1	1	yes	characteristic
69	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
70	C₂×C₇×C₈	SD₁₆	1	1	yes	characteristic
71	C₂×C₇×C₈	C₂×D₄	1	1	yes	characteristic
72	C₄⋊Dic₇	—	4	2	no	—
73	C₄⋊Dic₇	—	8	1	no	—
74	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
75	C₇×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
76	C₇×C₁₆	C₂²×C₄	1	1	yes	characteristic
77	C₇×C₁₆	C₂×D₄	1	1	yes	characteristic
78	C₂×#135	—	2	1	no	—
79	#508	—	2	1	no	—
80	#508	—	2	1	no	—
81	#508	Dic₇	1	2	yes	complemented
82	#508	C₂×D₇	1	2	yes	—
83	C₂×Q₃₂	Dic₇	1	1	yes	characteristic, complemented
84	C₂×SD₃₂	—	2	1	no	—
85	C₂×D₁₆	Dic₇	1	1	yes	characteristic, complemented
86	C₂²×C₁₆	C₂×D₇	1	1	yes	characteristic
87	C₂×#107	—	14	1	no	—
88	M₆₄	—	7	2	no	—
89	M₆₄	—	14	1	no	—
90	C₂×C₃₂	—	14	1	no	—
91	#367	—	7	2	no	—
92	#367	—	14	1	no	—
93	C₂³×C₇	—	8	1	no	—
94	C₂³×C₇	D₁₆	1	1	yes	characteristic
95	C₇×Q₈	—	4	2	no	—
96	C₇×Q₈	—	8	1	no	—
97	C₇×D₄	—	4	2	no	—
98	C₇×D₄	—	8	2	no	—
99	C₇×D₄	—	8	2	no	—
100	C₇×D₄	—	8	1	no	—
101	C₂×C₄×C₇	—	8	1	no	—
102	C₂×C₄×C₇	—	8	2	no	—
103	C₂×C₄×C₇	—	8	2	no	—
104	C₂×C₄×C₇	#102	1	1	yes	characteristic
105	C₂×C₄×C₇	#102	1	1	yes	characteristic
106	C₂×C₄×C₇	D₁₆	1	1	yes	characteristic
107	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
108	C₂×C₄×C₇	SD₃₂	1	1	yes	characteristic
109	C₂×C₄×C₇	C₂×D₈	1	1	yes	characteristic
110	C₂×Dic₇	—	8	2	no	—
111	C₂×Dic₇	—	16	2	no	—
112	C₇×C₈	#102	1	1	yes	characteristic
113	C₇×C₈	#102	1	1	yes	characteristic
114	C₇×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₇×C₈	C₂×SD₁₆	1	1	yes	characteristic
116	C₂×#41	—	4	1	no	—
117	#135	—	2	2	no	—
118	#135	—	2	2	no	—
119	#135	—	4	1	no	—
120	#135	—	4	1	no	—
121	C₂×Q₁₆	—	2	1	no	—
122	C₂×SD₁₆	—	4	1	no	—
123	C₂×D₈	—	2	1	no	—
124	C₂²×C₈	C₂²⋊D₇	1	1	yes	characteristic
125	C₂×C₄⋊C₄	—	28	1	no	—
126	Q₃₂	—	2	1	no	—
127	Q₃₂	C₂×Dic₇	1	2	yes	complemented
128	SD₃₂	—	2	2	no	—
129	SD₃₂	—	2	2	no	—
130	D₁₆	—	2	1	no	—
131	D₁₆	C₂×Dic₇	1	2	yes	complemented
132	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
133	C₂×C₁₆	C₂×Dic₇	1	1	yes	characteristic
134	C₂×C₁₆	C₂²⋊D₇	1	1	yes	characteristic
135	C₂×C₁₆	C₂²⋊D₇	1	1	yes	characteristic
136	C₂×C₁₆	C₂²⋊D₇	1	1	yes	characteristic
137	C₂×C₁₆	C₂²×D₇	1	1	yes	characteristic
138	#107	—	14	2	no	—
139	#107	—	28	1	no	—
140	C₃₂	—	14	2	no	—
141	C₂²×C₇	—	2	2	no	—
142	C₂²×C₇	—	8	2	no	—
143	C₂²×C₇	—	16	2	no	—
144	C₂²×C₇	#357	1	1	yes	characteristic
145	C₂²×C₇	#357	1	1	yes	characteristic
146	C₂²×C₇	C₂×D₁₆	1	1	yes	characteristic
147	C₄×C₇	—	8	2	no	—
148	C₄×C₇	#357	1	1	yes	characteristic
149	C₄×C₇	#357	1	1	yes	characteristic
150	C₄×C₇	C₂×#102	1	1	yes	characteristic
151	C₄×C₇	C₂×SD₃₂	1	1	yes	characteristic
152	Dic₇	—	16	2	no	—
153	#41	—	4	2	no	—
154	#41	—	4	2	no	—
155	#41	—	8	1	no	—
156	#41	—	8	1	no	—
157	C₂×Q₈	—	4	1	no	—
158	C₂×D₄	—	4	1	no	—
159	C₂×D₄	—	8	1	no	—
160	C₂²×C₄	—	8	1	no	—
161	C₂²×C₄	—	56	1	no	—
162	C₂²×C₄	#1141	1	1	yes	characteristic
163	Q₁₆	—	2	2	no	—
164	Q₁₆	—	4	1	no	—
165	SD₁₆	—	4	2	no	—
166	SD₁₆	—	4	2	no	—
167	D₈	—	2	2	no	—
168	D₈	—	4	1	no	—
169	C₂×C₈	#1141	1	1	yes	characteristic
170	C₂×C₈	#1142	1	1	yes	characteristic
171	C₂×C₈	#1142	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
173	C₂×C₈	C₂²⋊Dic₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₇	1	1	yes	characteristic
175	C₄⋊C₄	—	28	2	no	—
176	C₄⋊C₄	—	56	1	no	—
177	C₁₆	C₂²⋊Dic₇	1	1	yes	characteristic
178	C₁₆	C₂²⋊Dic₇	1	1	yes	characteristic
179	C₁₆	C₂²×Dic₇	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₇	1	1	yes	characteristic
181	C₂×C₇	—	2	1	no	—
182	C₂×C₇	—	2	1	no	—
183	C₂×C₇	—	16	2	no	—
184	C₂×C₇	#1418	1	2	yes	complemented
185	C₂×C₇	C₂×#357	1	1	yes	characteristic
186	C₂³	—	8	1	no	—
187	C₂³	#6400	1	1	yes	characteristic
188	Q₈	—	4	2	no	—
189	Q₈	—	8	1	no	—
190	D₄	—	4	2	no	—
191	D₄	—	8	2	no	—
192	D₄	—	8	2	no	—
193	D₄	—	8	1	no	—
194	C₂×C₄	—	8	1	no	—
195	C₂×C₄	—	8	2	no	—
196	C₂×C₄	—	8	2	no	—
197	C₂×C₄	—	56	2	no	—
198	C₂×C₄	—	112	2	no	—
199	C₂×C₄	#6400	1	1	yes	characteristic
200	C₂×C₄	#6401	1	1	yes	characteristic
201	C₂×C₄	#6401	1	1	yes	characteristic
202	C₂×C₄	#6406	1	1	yes	characteristic
203	C₂×C₄	#6406	1	1	yes	characteristic
204	C₂×C₄	C₂×#1141	1	1	yes	characteristic
205	C₈	#6406	1	1	yes	characteristic
206	C₈	#6406	1	1	yes	characteristic
207	C₈	C₂×#1142	1	1	yes	characteristic
208	C₈	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
209	C₇	C₂×#1418	1	1	yes	characteristic, complemented, Sylow
210	C₂²	—	2	2	no	—
211	C₂²	—	8	2	no	—
212	C₂²	—	16	2	no	—
213	C₂²	#89357	1	1	yes	characteristic
214	C₂²	#89357	1	1	yes	characteristic
215	C₂²	C₂×#6400	1	1	yes	characteristic
216	C₄	—	8	2	no	—
217	C₄	—	112	2	no	—
218	C₄	#89357	1	1	yes	characteristic
219	C₄	#89357	1	1	yes	characteristic
220	C₄	C₂×#6401	1	1	yes	characteristic
221	C₄	C₂×#6406	1	1	yes	characteristic
222	C₂	—	2	1	no	—
223	C₂	—	2	1	no	—
224	C₂	—	16	2	no	—
225	C₂	#11725634	1	2	yes	direct product factor
226	C₂	C₂×#89357	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#89357
commutator	C₇×C₁₆	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₇
Fitting, prime radical	C₂×C₇×#508	C₂
socle	C₂²×C₇	#357
cosocle	C₂×C₇×C₁₆	C₂³
hypercenter	C₂×#508	D₇
hypocenter	C₇	C₂×#1418
LCS	G ▷ C₇×C₁₆ ▷ C₇×C₈ ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
2. C₂×#89300	C₂	semidirect
3. C₂×#89294	C₂	semidirect
4. #11725634	C₂	direct (N × H)
8. C₇×#508	C₄	semidirect
10. C₂×C₇×Q₃₂	C₄	semidirect
12. C₂×C₇×D₁₆	C₄	semidirect
16. #89300	C₂²	semidirect
18. #89294	C₂²	semidirect
32. C₇×Q₃₂	C₂×C₄	semidirect
36. C₇×D₁₆	C₂×C₄	semidirect
81. #508	Dic₇	semidirect
83. C₂×Q₃₂	Dic₇	semidirect
85. C₂×D₁₆	Dic₇	semidirect
127. Q₃₂	C₂×Dic₇	semidirect
131. D₁₆	C₂×Dic₇	semidirect
184. C₂×C₇	#1418	semidirect
209. C₇	C₂×#1418	semidirect
225. C₂	#11725634	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×#508	2
C₂×#89300	2
C₂×#89294	2
#11725634	2
C₂×#1418	7

Subgroups — C2×#11725634 #13274255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725634 #13274255