Subgroups — C₂×#11725843 #13323561

Order 1792.

654 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×Dic₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#89451	C₂	1	2	yes	maximal
3	C₂×C₇×#322	C₂	1	1	yes	maximal, characteristic
4	C₂×#89251	C₂	1	2	yes	maximal
5	C₂×#89252	C₂	1	1	yes	maximal, characteristic
6	#11725843	C₂	1	8	yes	maximal, direct product factor
7	C₂²×Dic₂₈	C₄	1	1	yes	characteristic
8	C₂×C₄×Dic₁₄	C₂²	1	2	yes	—
9	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
12	C₂×#6392	C₄	1	1	yes	characteristic
13	C₂×#6390	C₄	1	2	yes	—
14	C₂×#6389	—	2	2	no	—
15	C₂×Dic₇⋊C₈	—	2	2	no	—
16	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	2	yes	—
17	C₄×Dic₂₈	C₄	1	4	yes	—
18	C₄×Dic₂₈	C₂²	1	4	yes	—
19	#89451	—	2	4	no	—
20	#89451	C₂²	1	8	yes	—
21	C₇×#322	C₂²	1	4	yes	—
22	#89252	C₂²	1	4	yes	—
23	#89251	C₂²	1	8	yes	—
24	C₂×#1593	—	7	1	no	maximal, Sylow
25	C₂²×Dic₁₄	C₂×C₄	1	2	yes	—
26	C₂²×C₇×C₈	—	2	1	no	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	D₄	1	2	yes	—
30	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₄⋊Dic₇	C₂×C₄	1	2	yes	—
32	C₂×Dic₇⋊C₄	—	2	2	no	—
33	C₂×C₄×Dic₇	—	2	2	no	—
34	C₂²×C₇⋊C₈	—	2	2	no	—
35	C₂×Dic₂₈	—	2	4	no	—
36	C₂×Dic₂₈	C₂×C₄	1	4	yes	—
37	C₄×Dic₁₄	—	2	2	no	—
38	C₄×Dic₁₄	—	2	2	no	—
39	C₄×Dic₁₄	C₂×C₄	1	4	yes	—
40	C₄×Dic₁₄	C₂³	1	4	yes	—
41	C₄×C₇×C₈	—	2	1	no	—
42	C₄×C₇×C₈	C₂×C₄	1	2	yes	—
43	C₄×C₇×C₈	D₄	1	2	yes	—
44	C₄×C₇×C₈	C₂³	1	2	yes	—
45	C₄×C₇×C₈	C₂³	1	2	yes	—
46	C₄×C₇×C₈	C₂³	1	2	yes	—
47	#6392	C₂×C₄	1	4	yes	—
48	#6390	C₂×C₄	1	8	yes	—
49	#6389	—	2	8	no	—
50	Dic₇⋊C₈	—	2	8	no	—
51	C₄⋊(C₇⋊C₈)	—	2	2	no	—
52	C₄⋊(C₇⋊C₈)	C₂³	1	4	yes	—
53	C₂×#446	—	7	2	no	—
54	C₂×C₄×Q₁₆	—	7	1	no	—
55	#1593	—	7	8	no	—
56	C₂×#334	—	7	1	no	—
57	C₂×Q₈⋊C₈	—	7	2	no	—
58	C₂×#322	D₇	1	1	yes	characteristic
59	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₇	#41	1	1	yes	characteristic
62	C₂²×Dic₇	—	2	2	no	—
63	C₂²×Dic₇	—	2	2	no	—
64	C₂×Dic₁₄	—	2	4	no	—
65	C₂×Dic₁₄	—	2	4	no	—
66	C₂×Dic₁₄	—	2	2	no	—
67	C₂×Dic₁₄	C₂²×C₄	1	4	yes	—
68	C₂×C₇×C₈	—	2	2	no	—
69	C₂×C₇×C₈	—	2	4	no	—
70	C₂×C₇×C₈	—	2	2	no	—
71	C₂×C₇×C₈	—	4	2	no	—
72	C₂×C₇×C₈	M₁₆	1	4	yes	—
73	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
74	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
75	C₂×C₇×C₈	C₂×D₄	1	4	yes	—
76	C₄²×C₇	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₇	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₇	#41	1	1	yes	characteristic
79	C₄²×C₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊Dic₇	—	2	2	no	—
81	C₄⋊Dic₇	C₂²×C₄	1	4	yes	—
82	Dic₇⋊C₄	—	2	4	no	—
83	Dic₇⋊C₄	—	2	4	no	—
84	C₄×Dic₇	—	2	4	no	—
85	C₄×Dic₇	—	2	4	no	—
86	C₂×C₇⋊C₈	—	2	4	no	—
87	C₂×C₇⋊C₈	—	4	4	no	—
88	Dic₂₈	—	2	8	no	—
89	C₂²×Q₁₆	—	7	1	no	—
90	C₂×C₄×Q₈	—	7	2	no	—
91	#446	—	7	8	no	—
92	#446	—	14	4	no	—
93	C₄×Q₁₆	—	7	4	no	—
94	C₄×Q₁₆	—	7	4	no	—
95	C₂×#107	—	7	1	no	—
96	C₂×C₄⋊C₈	—	7	2	no	—
97	C₂×C₄⋊C₈	—	14	2	no	—
98	C₂×#103	—	7	2	no	—
99	C₂×#97	—	14	2	no	—
100	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
101	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
102	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
103	#334	—	7	4	no	—
104	Q₈⋊C₈	—	7	8	no	—
105	#322	C₂×D₇	1	4	yes	—
106	C₂³×C₇	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₇	C₂×M₁₆	1	2	yes	—
112	C₂×C₄×C₇	#131	1	2	yes	—
113	C₂×C₄×C₇	#137	1	4	yes	—
114	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
117	C₂×Dic₇	—	2	4	no	—
118	C₂×Dic₇	—	2	4	no	—
119	C₂×Dic₇	—	4	4	no	—
120	C₂×Dic₇	—	4	4	no	—
121	Dic₁₄	—	2	4	no	—
122	Dic₁₄	—	2	4	no	—
123	Dic₁₄	—	4	4	no	—
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	—	2	4	no	—
126	C₇×C₈	—	4	2	no	—
127	C₇×C₈	C₂×M₁₆	1	4	yes	—
128	C₇⋊C₈	—	4	4	no	—
129	C₂²×Q₈	—	7	2	no	—
130	C₂×Q₁₆	—	7	4	no	—
131	C₂×Q₁₆	—	14	4	no	—
132	C₂²×C₈	—	2	1	no	—
133	C₂²×C₈	—	14	2	no	—
134	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
135	C₂²×C₈	C₄×D₇	1	1	yes	characteristic
136	C₂²×C₈	D₂₈	1	2	yes	—
137	C₄×Q₈	—	7	4	no	—
138	C₄×Q₈	—	7	4	no	—
139	C₄×Q₈	—	14	2	no	—
140	C₄×Q₈	—	14	2	no	—
141	C₂×C₄⋊C₄	—	7	2	no	—
142	C₂×C₄⋊C₄	—	14	2	no	—
143	C₂×C₄²	—	14	2	no	—
144	C₂×C₄²	C₂²×D₇	1	1	yes	characteristic
145	#107	—	7	4	no	—
146	C₄⋊C₈	—	7	4	no	—
147	C₄⋊C₈	—	14	8	no	—
148	C₄⋊C₈	—	14	2	no	—
149	#103	—	7	8	no	—
150	#97	—	14	8	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	C₄×D₇	1	2	yes	—
153	C₄×C₈	D₂₈	1	2	yes	—
154	C₄×C₈	C₂²×D₇	1	2	yes	—
155	C₄×C₈	C₂²×D₇	1	2	yes	—
156	C₄×C₈	C₂²×D₇	1	2	yes	—
157	C₂²×C₇	#436	1	2	yes	—
158	C₂²×C₇	#441	1	2	yes	—
159	C₂²×C₇	#444	1	2	yes	—
160	C₂²×C₇	C₂×C₄×D₄	1	1	yes	characteristic
161	C₄×C₇	—	2	1	no	—
162	C₄×C₇	—	2	1	no	—
163	C₄×C₇	#436	1	2	yes	—
164	C₄×C₇	#441	1	2	yes	—
165	C₄×C₇	C₂²×M₁₆	1	1	yes	characteristic
166	C₄×C₇	C₂×#131	1	1	yes	characteristic
167	C₄×C₇	C₂×#137	1	2	yes	—
168	Dic₇	—	4	4	no	—
169	Dic₇	—	4	4	no	—
170	C₂×Q₈	—	7	4	no	—
171	C₂×Q₈	—	14	2	no	—
172	C₂×Q₈	—	14	4	no	—
173	C₂×Q₈	—	14	4	no	—
174	C₂²×C₄	—	14	2	no	—
175	C₂²×C₄	—	14	2	no	—
176	C₂²×C₄	C₂×C₄×D₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₂₈	1	1	yes	characteristic
178	C₂²×C₄	#1157	1	1	yes	characteristic
179	Q₁₆	—	14	8	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	2	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	14	4	no	—
185	C₂×C₈	—	28	4	no	—
186	C₂×C₈	#1131	1	4	yes	—
187	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
188	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
189	C₂×C₈	C₂×D₂₈	1	4	yes	—
190	C₄⋊C₄	—	7	4	no	—
191	C₄⋊C₄	—	14	4	no	—
192	C₄⋊C₄	—	14	2	no	—
193	C₄⋊C₄	—	14	4	no	—
194	C₄²	—	14	4	no	—
195	C₄²	—	14	4	no	—
196	C₄²	C₂×C₄×D₇	1	1	yes	characteristic
197	C₄²	C₂×D₂₈	1	1	yes	characteristic
198	C₄²	#1157	1	1	yes	characteristic
199	C₄²	C₂³×D₇	1	1	yes	characteristic
200	C₂×C₇	#1593	1	4	yes	complemented
201	C₂×C₇	C₂×#436	1	1	yes	characteristic
202	C₂×C₇	C₂×#441	1	1	yes	characteristic
203	C₂×C₇	C₂×#444	1	1	yes	characteristic
204	C₂³	C₄×D₂₈	1	1	yes	characteristic
205	Q₈	—	14	4	no	—
206	Q₈	—	14	4	no	—
207	Q₈	—	28	4	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	14	4	no	—
210	C₂×C₄	—	14	4	no	—
211	C₂×C₄	—	28	4	no	—
212	C₂×C₄	—	28	4	no	—
213	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
214	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
215	C₂×C₄	C₄×D₂₈	1	1	yes	characteristic
216	C₂×C₄	C₂×#1131	1	2	yes	—
217	C₂×C₄	#6464	1	2	yes	—
218	C₂×C₄	#6472	1	4	yes	—
219	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
220	C₂×C₄	C₂²×D₂₈	1	1	yes	characteristic
221	C₂×C₄	C₂×#1157	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	2	4	no	—
224	C₈	—	4	2	no	—
225	C₈	—	28	4	no	—
226	C₈	C₂×#1131	1	4	yes	—
227	C₇	C₂×#1593	1	1	yes	characteristic, complemented, Sylow
228	C₂²	#89460	1	2	yes	—
229	C₂²	#89488	1	2	yes	—
230	C₂²	#89489	1	2	yes	—
231	C₂²	C₂×C₄×D₂₈	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	28	4	no	—
235	C₄	—	28	4	no	—
236	C₄	#89460	1	2	yes	—
237	C₄	#89488	1	2	yes	—
238	C₄	C₂²×#1131	1	1	yes	characteristic
239	C₄	C₂×#6464	1	1	yes	characteristic
240	C₄	C₂×#6472	1	2	yes	—
241	C₂	#11725843	1	4	yes	direct product factor
242	C₂	C₂×#89460	1	1	yes	characteristic
243	C₂	C₂×#89488	1	1	yes	characteristic
244	C₂	C₂×#89489	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₈
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₄²	C₂³×D₇
Fitting	C₂×C₇×#322	C₂
socle, prime radical	C₂³×C₇	C₄×D₄
cosocle	C₄²×C₇	C₂⁴
hypercenter	C₂×#322	D₇
hypocenter	C₇	C₂×#1593
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#322

Complements

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

N	H ≅ G/N	type
6. #11725843	C₂	direct (N × H)
200. C₂×C₇	#1593	semidirect
227. C₇	C₂×#1593	semidirect
241. C₂	#11725843	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×Dic₂₈	2
C₂×#89451	2
C₂×C₇×#322	2
C₂×#89251	2
C₂×#89252	2
#11725843	2
C₂×#1593	7

Subgroups — C2×#11725843 #13323561

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11725843 #13323561