Subgroups — C₂×#11727603 #13319502

Order 1792.

994 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₇×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#1136	C₂	1	2	yes	maximal
3	#11727603	C₂	1	8	yes	maximal, direct product factor
4	C₂×#89250	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#89249	C₂	1	2	yes	maximal, complemented
6	C₂×C₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₇×M₁₆	C₄	1	2	yes	—
8	C₂²×C₄²×C₇	C₂²	1	1	yes	characteristic
9	C₂²×#1136	C₄	1	4	yes	—
10	C₄×C₇×M₁₆	C₄	1	4	yes	—
11	C₄×C₇×M₁₆	C₂²	1	4	yes	—
12	C₂×C₇×#97	C₄	1	2	yes	—
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
15	C₄×#1136	C₄	1	8	yes	—
16	C₄×#1136	C₂²	1	8	yes	—
17	C₂×#6377	C₄	1	4	yes	—
18	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
19	C₂×C₄×C₇⋊C₈	C₂²	1	2	yes	—
20	#89250	C₂²	1	4	yes	complemented
21	#89249	C₂²	1	8	yes	complemented
22	C₈×C₇⋊C₈	C₂²	1	4	yes	complemented
23	C₂×#1455	—	7	1	no	maximal, Sylow
24	C₂³×C₄×C₇	C₂×C₄	1	2	yes	—
25	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₇×M₁₆	—	2	8	no	—
27	C₂×C₇×M₁₆	C₂×C₄	1	8	yes	—
28	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
29	C₂²×C₇×C₈	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	4	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₇	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₇	C₂³	1	2	yes	—
35	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
36	C₂×#1136	—	2	16	no	—
37	C₂×#1136	C₂×C₄	1	16	yes	—
38	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₂²×C₇⋊C₈	C₂×C₄	1	4	yes	—
40	C₇×#97	C₂×C₄	1	8	yes	—
41	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
42	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₇×C₈	C₂³	1	2	yes	—
44	C₄×C₇×C₈	C₂³	1	2	yes	—
45	#6377	C₂×C₄	1	16	yes	—
46	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
47	C₄×C₇⋊C₈	C₂×C₄	1	4	yes	—
48	C₄×C₇⋊C₈	C₂³	1	4	yes	—
49	C₄×C₇⋊C₈	C₂³	1	4	yes	—
50	C₂×C₄×M₁₆	—	7	2	no	—
51	C₂×C₄×M₁₆	D₇	1	1	yes	characteristic
52	#1455	—	7	8	no	—
53	C₂×#322	—	7	2	no	—
54	C₂×#322	—	7	1	no	—
55	C₂×C₈²	—	7	1	no	—
56	C₂⁴×C₇	C₄²	1	1	yes	characteristic
57	C₂²×C₄×C₇	—	2	8	no	—
58	C₂²×C₄×C₇	—	2	4	no	—
59	C₂²×C₄×C₇	C₄²	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₇	C₄²	1	2	yes	—
62	C₂²×C₄×C₇	C₄²	1	4	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₇	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
67	C₇×M₁₆	—	2	16	no	—
68	C₂×C₇×C₈	—	2	4	no	—
69	C₂×C₇×C₈	M₁₆	1	8	yes	complemented
70	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₇×C₈	C₂²×C₄	1	4	yes	—
72	C₄²×C₇	C₄²	1	4	yes	—
73	C₄²×C₇	C₄²	1	2	yes	—
74	C₄²×C₇	C₂²×C₄	1	2	yes	—
75	C₄²×C₇	C₂²×C₄	1	4	yes	—
76	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂²×C₄	1	2	yes	—
78	C₄²×C₇	C₂⁴	1	1	yes	characteristic
79	#1136	—	2	32	no	—
80	C₂×C₇⋊C₈	—	2	8	no	—
81	C₂×C₇⋊C₈	M₁₆	1	16	yes	complemented
82	C₂×C₇⋊C₈	C₂²×C₄	1	8	yes	—
83	C₂×C₇⋊C₈	C₂²×C₄	1	8	yes	—
84	C₂²×M₁₆	—	7	4	no	—
85	C₂²×M₁₆	Dic₇	1	2	yes	—
86	C₂²×C₄²	C₂×D₇	1	1	yes	characteristic
87	C₄×M₁₆	—	7	8	no	—
88	C₄×M₁₆	—	7	8	no	—
89	C₄×M₁₆	Dic₇	1	4	yes	—
90	C₄×M₁₆	C₂×D₇	1	4	yes	—
91	C₂×#97	—	7	4	no	—
92	C₂×#97	Dic₇	1	2	yes	—
93	C₂×C₄×C₈	—	7	2	no	—
94	C₂×C₄×C₈	—	7	2	no	—
95	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
96	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
97	#322	—	7	8	no	—
98	#322	—	7	4	no	—
99	C₈²	—	7	4	no	—
100	C₂³×C₇	—	2	4	no	—
101	C₂³×C₇	—	2	2	no	—
102	C₂³×C₇	C₂×C₄²	1	2	yes	—
103	C₂³×C₇	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇	—	2	4	no	—
105	C₂×C₄×C₇	—	2	2	no	—
106	C₂×C₄×C₇	—	2	8	no	—
107	C₂×C₄×C₇	—	2	4	no	—
108	C₂×C₄×C₇	—	2	8	no	—
109	C₂×C₄×C₇	—	2	4	no	—
110	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₇	C₂×C₄²	1	4	yes	—
112	C₂×C₄×C₇	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂×M₁₆	1	8	yes	—
115	C₂×C₄×C₇	C₂×M₁₆	1	4	yes	—
116	C₂×C₄×C₇	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
118	C₇×C₈	—	2	4	no	—
119	C₇×C₈	C₂×M₁₆	1	8	yes	complemented
120	C₇⋊C₈	—	2	8	no	—
121	C₇⋊C₈	C₂×M₁₆	1	16	yes	complemented
122	C₂³×C₄	C₄×D₇	1	2	yes	—
123	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
124	C₂×M₁₆	—	2	8	no	—
125	C₂×M₁₆	—	7	16	no	—
126	C₂×M₁₆	—	14	16	no	—
127	C₂×M₁₆	C₂×Dic₇	1	8	yes	—
128	C₂²×C₈	—	7	4	no	—
129	C₂²×C₈	—	7	4	no	—
130	C₂²×C₈	C₂×Dic₇	1	2	yes	—
131	C₂²×C₈	C₂×Dic₇	1	2	yes	—
132	C₂×C₄²	C₄×D₇	1	2	yes	—
133	C₂×C₄²	C₄×D₇	1	4	yes	—
134	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
135	C₂×C₄²	C₂×Dic₇	1	2	yes	—
136	C₂×C₄²	C₂²×D₇	1	2	yes	—
137	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
138	#97	—	7	16	no	—
139	#97	C₂×Dic₇	1	8	yes	—
140	C₄×C₈	—	7	4	no	—
141	C₄×C₈	—	7	4	no	—
142	C₄×C₈	—	7	4	no	—
143	C₄×C₈	—	7	4	no	—
144	C₄×C₈	C₂×Dic₇	1	2	yes	—
145	C₄×C₈	C₂×Dic₇	1	2	yes	—
146	C₄×C₈	C₂²×D₇	1	2	yes	—
147	C₄×C₈	C₂²×D₇	1	2	yes	—
148	C₂²×C₇	—	2	4	no	—
149	C₂²×C₇	—	2	2	no	—
150	C₂²×C₇	—	4	4	no	—
151	C₂²×C₇	C₄×M₁₆	1	4	yes	—
152	C₂²×C₇	C₄×M₁₆	1	2	yes	—
153	C₂²×C₇	C₂²×C₄²	1	1	yes	characteristic
154	C₄×C₇	—	2	4	no	—
155	C₄×C₇	—	2	2	no	—
156	C₄×C₇	C₄×M₁₆	1	4	yes	—
157	C₄×C₇	C₄×M₁₆	1	2	yes	—
158	C₄×C₇	C₂²×M₁₆	1	4	yes	—
159	C₄×C₇	C₂²×M₁₆	1	2	yes	—
160	C₂⁴	C₄×Dic₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	8	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	C₄×Dic₇	1	1	yes	characteristic
164	C₂²×C₄	C₄×Dic₇	1	2	yes	—
165	C₂²×C₄	C₄×Dic₇	1	2	yes	—
166	C₂²×C₄	C₄×Dic₇	1	4	yes	—
167	C₂²×C₄	C₂×C₄×D₇	1	4	yes	—
168	C₂²×C₄	C₂×C₄×D₇	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₇	1	2	yes	—
170	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
171	M₁₆	—	2	16	no	—
172	M₁₆	—	14	32	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	7	16	no	—
175	C₂×C₈	—	7	8	no	—
176	C₂×C₈	—	7	8	no	—
177	C₂×C₈	—	14	8	no	—
178	C₂×C₈	#1136	1	8	yes	complemented
179	C₂×C₈	C₂²×Dic₇	1	4	yes	—
180	C₂×C₈	C₂²×Dic₇	1	4	yes	—
181	C₄²	C₄×Dic₇	1	4	yes	—
182	C₄²	C₄×Dic₇	1	2	yes	—
183	C₄²	C₂×C₄×D₇	1	2	yes	—
184	C₄²	C₂×C₄×D₇	1	4	yes	—
185	C₄²	C₂²×Dic₇	1	1	yes	characteristic
186	C₄²	C₂²×Dic₇	1	2	yes	—
187	C₄²	C₂³×D₇	1	1	yes	characteristic
188	C₂×C₇	—	4	2	no	—
189	C₂×C₇	#1455	1	4	yes	complemented
190	C₂×C₇	C₂×C₄×M₁₆	1	2	yes	—
191	C₂×C₇	C₂×C₄×M₁₆	1	1	yes	characteristic
192	C₂³	—	2	4	no	—
193	C₂³	—	2	2	no	—
194	C₂³	C₂×C₄×Dic₇	1	2	yes	—
195	C₂³	C₂×C₄×Dic₇	1	1	yes	characteristic
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	8	no	—
199	C₂×C₄	—	2	8	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	D₇×M₁₆	1	8	yes	—
203	C₂×C₄	C₂×#1136	1	4	yes	—
204	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
205	C₂×C₄	C₂×C₄×Dic₇	1	4	yes	—
206	C₂×C₄	C₂×C₄×Dic₇	1	2	yes	—
207	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
208	C₂×C₄	C₂²×C₄×D₇	1	2	yes	—
209	C₂×C₄	C₂³×Dic₇	1	1	yes	characteristic
210	C₈	—	2	4	no	—
211	C₈	—	7	16	no	—
212	C₈	—	14	8	no	—
213	C₈	C₂×#1136	1	8	yes	complemented
214	C₇	C₂×#1455	1	1	yes	characteristic, complemented, Sylow
215	C₂²	—	2	4	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	C₄×#1136	1	2	yes	—
219	C₂²	M₁₆×Dic₇	1	4	yes	—
220	C₂²	C₂²×C₄×Dic₇	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	—	2	2	no	—
223	C₄	C₄×#1136	1	2	yes	—
224	C₄	M₁₆×Dic₇	1	4	yes	—
225	C₄	C₂×D₇×M₁₆	1	4	yes	—
226	C₄	C₂²×#1136	1	2	yes	—
227	C₂	—	4	2	no	—
228	C₂	#11727603	1	4	yes	direct product factor
229	C₂	C₂×C₄×#1136	1	1	yes	characteristic
230	C₂	C₂×M₁₆×Dic₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₇
commutator	C₂²×C₇	C₂²×C₄²
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₄×C₇×M₁₆	C₂
socle	C₂³×C₇	C₂×C₄²
prime radical	C₂⁴×C₇	C₄²
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₇
hypocenter	C₇	C₂×#1455
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #11727603	C₂	direct (N × H)
4. C₂×#89250	C₂	semidirect
5. C₂×#89249	C₂	semidirect
6. C₂×C₈×C₇⋊C₈	C₂	semidirect
20. #89250	C₂²	semidirect
21. #89249	C₂²	semidirect
22. C₈×C₇⋊C₈	C₂²	semidirect
69. C₂×C₇×C₈	M₁₆	semidirect
81. C₂×C₇⋊C₈	M₁₆	semidirect
119. C₇×C₈	C₂×M₁₆	semidirect
121. C₇⋊C₈	C₂×M₁₆	semidirect
178. C₂×C₈	#1136	semidirect
189. C₂×C₇	#1455	semidirect
213. C₈	C₂×#1136	semidirect
214. C₇	C₂×#1455	semidirect
228. C₂	#11727603	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×C₄×C₇×M₁₆	2
C₂×C₄×#1136	2
#11727603	2
C₂×#89250	2
C₂×#89249	2
C₂×C₈×C₇⋊C₈	2
C₂×#1455	7

Subgroups — C2×#11727603 #13319502

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11727603 #13319502