Subgroups — C₄×#6451 #11731069

Order 896.

588 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#6451	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₄	C₂	1	2	yes	maximal
3	C₄×C₇×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	#89757	C₂	1	1	yes	maximal, characteristic
5	#89748	C₂	1	1	yes	maximal, characteristic
6	#89745	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊Dic₇	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₇⋊C₄	C₂	1	2	yes	maximal
9	C₄²×Dic₇	C₂	1	1	yes	maximal, characteristic
10	#89419	C₂	1	2	yes	maximal
11	#89418	C₂	1	2	yes	maximal
12	C₂²×Dic₁₄	C₂²	1	2	yes	—
13	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂×C₇×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
21	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
23	C₂×Dic₇⋊C₄	C₂²	1	2	yes	—
24	C₂×C₄×Dic₇	C₂²	1	2	yes	—
25	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Dic₇	C₂²	1	1	yes	characteristic
31	#6451	C₄	1	8	yes	direct product factor
32	C₄×Dic₁₄	—	2	8	no	—
33	C₇×#95	C₂²	1	2	yes	—
34	C₇⋊#95	C₂²	1	2	yes	—
35	C₇⋊#95	C₂²	1	2	yes	—
36	C₇⋊#95	C₂²	1	2	yes	—
37	C₄×C₄⋊Q₈	—	7	1	no	maximal, Sylow
38	C₂²×C₄×C₇	C₂³	1	2	yes	—
39	C₂²×C₄×C₇	C₂³	1	2	yes	—
40	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₇	C₂³	1	2	yes	—
44	C₂²×Dic₇	C₂³	1	2	yes	—
45	C₂²×Dic₇	C₂³	1	1	yes	characteristic
46	C₂²×Dic₇	C₂³	1	1	yes	characteristic
47	C₂²×Dic₇	C₂³	1	1	yes	characteristic
48	C₂²×Dic₇	C₂³	1	1	yes	characteristic
49	C₂×Dic₁₄	—	2	4	no	—
50	C₂×Dic₁₄	—	2	4	no	—
51	C₂×Dic₁₄	C₂×C₄	1	8	yes	—
52	C₇×C₄⋊C₄	—	2	2	no	—
53	C₇×C₄⋊C₄	C₂×C₄	1	4	yes	—
54	C₄²×C₇	—	2	4	no	—
55	C₄²×C₇	Q₈	1	4	yes	complemented
56	C₄⋊Dic₇	—	2	2	no	—
57	C₄⋊Dic₇	—	2	4	no	—
58	C₄⋊Dic₇	C₂×C₄	1	4	yes	—
59	Dic₇⋊C₄	—	2	4	no	—
60	Dic₇⋊C₄	—	2	4	no	—
61	Dic₇⋊C₄	—	2	4	no	—
62	Dic₇⋊C₄	C₂×C₄	1	8	yes	—
63	C₄×Dic₇	—	2	4	no	—
64	C₄×Dic₇	—	2	2	no	—
65	C₄×Dic₇	—	2	2	no	—
66	C₄×Dic₇	—	2	2	no	—
67	C₄×Dic₇	C₂×C₄	1	4	yes	—
68	C₄×Dic₇	D₄	1	4	yes	—
69	C₄×Dic₇	Q₈	1	4	yes	complemented
70	C₂×C₄⋊Q₈	—	7	1	no	—
71	C₂×C₄×Q₈	—	7	2	no	—
72	#391	—	7	2	no	—
73	#389	—	7	1	no	—
74	#389	—	7	1	no	—
75	#389	—	7	2	no	—
76	#384	—	7	1	no	—
77	C₄×C₄⋊C₄	—	7	2	no	—
78	C₄×C₄⋊C₄	—	7	1	no	—
79	C₄×C₄⋊C₄	D₇	1	1	yes	characteristic
80	C₄³	—	7	1	no	—
81	C₂³×C₇	C₂⁴	1	1	yes	characteristic
82	C₂×C₄×C₇	—	2	2	no	—
83	C₂×C₄×C₇	—	2	1	no	—
84	C₂×C₄×C₇	—	2	1	no	—
85	C₂×C₄×C₇	—	2	2	no	—
86	C₂×C₄×C₇	—	2	2	no	—
87	C₂×C₄×C₇	—	2	2	no	—
88	C₂×C₄×C₇	—	2	2	no	—
89	C₂×C₄×C₇	C₂²×C₄	1	4	yes	—
90	C₂×C₄×C₇	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₇	C₂×D₄	1	2	yes	—
92	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
93	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
94	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
95	C₂×C₄×C₇	C₂×Q₈	1	2	yes	—
96	C₂×C₄×C₇	#41	1	2	yes	—
97	C₂×Dic₇	—	2	2	no	—
98	C₂×Dic₇	—	2	1	no	—
99	C₂×Dic₇	—	2	1	no	—
100	C₂×Dic₇	—	2	1	no	—
101	C₂×Dic₇	—	2	1	no	—
102	C₂×Dic₇	—	2	2	no	—
103	C₂×Dic₇	—	2	2	no	—
104	C₂×Dic₇	—	2	2	no	—
105	C₂×Dic₇	—	2	2	no	—
106	C₂×Dic₇	C₂²×C₄	1	4	yes	—
107	C₂×Dic₇	C₂²×C₄	1	2	yes	—
108	C₂×Dic₇	C₂²×C₄	1	2	yes	—
109	C₂×Dic₇	C₂×D₄	1	2	yes	—
110	C₂×Dic₇	C₂×D₄	1	2	yes	—
111	C₂×Dic₇	C₂×Q₈	1	2	yes	—
112	C₂×Dic₇	C₂×Q₈	1	2	yes	—
113	C₂×Dic₇	#41	1	2	yes	—
114	C₂×Dic₇	#41	1	2	yes	—
115	Dic₁₄	—	2	16	no	—
116	C₂²×Q₈	—	7	2	no	—
117	C₄⋊Q₈	—	7	8	no	—
118	C₄×Q₈	—	14	8	no	—
119	C₂×C₄⋊C₄	—	7	2	no	—
120	C₂×C₄⋊C₄	—	7	1	no	—
121	C₂×C₄⋊C₄	—	7	2	no	—
122	C₂×C₄⋊C₄	—	7	2	no	—
123	C₂×C₄⋊C₄	—	7	2	no	—
124	C₂×C₄⋊C₄	—	7	2	no	—
125	C₂×C₄⋊C₄	—	7	1	no	—
126	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
127	C₂×C₄⋊C₄	C₂×D₇	1	1	yes	characteristic
128	C₂×C₄²	—	7	1	no	—
129	C₂×C₄²	—	7	1	no	—
130	C₂×C₄²	—	7	1	no	—
131	C₂×C₄²	—	7	1	no	—
132	C₂×C₄²	—	7	1	no	—
133	C₂×C₄²	—	7	1	no	—
134	C₂×C₄²	—	7	2	no	—
135	C₂×C₄²	C₂×D₇	1	2	yes	—
136	C₂×C₄²	C₂×D₇	1	1	yes	characteristic
137	#95	—	7	2	no	—
138	#95	—	7	2	no	—
139	#95	—	7	2	no	—
140	#95	C₂×D₇	1	2	yes	—
141	C₂²×C₇	C₂³×C₄	1	1	yes	characteristic
142	C₂²×C₇	C₂²×D₄	1	1	yes	characteristic
143	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
144	C₂²×C₇	C₂²×Q₈	1	1	yes	characteristic
145	C₂²×C₇	C₂×#41	1	1	yes	characteristic
146	C₂²×C₇	C₂×#41	1	1	yes	characteristic
147	C₂²×C₇	C₂×#41	1	1	yes	characteristic
148	C₄×C₇	—	2	4	no	—
149	C₄×C₇	—	2	2	no	—
150	C₄×C₇	—	4	2	no	—
151	C₄×C₇	C₄×Q₈	1	4	yes	complemented
152	C₄×C₇	C₄⋊Q₈	1	4	yes	complemented
153	Dic₇	—	2	4	no	—
154	Dic₇	—	2	2	no	—
155	Dic₇	—	2	2	no	—
156	Dic₇	—	4	2	no	—
157	Dic₇	C₄×D₄	1	4	yes	—
158	Dic₇	C₄×Q₈	1	4	yes	complemented
159	C₂×Q₈	—	7	8	no	—
160	C₂×Q₈	—	14	4	no	—
161	C₂×Q₈	—	14	4	no	—
162	C₂²×C₄	—	7	1	no	—
163	C₂²×C₄	—	7	1	no	—
164	C₂²×C₄	—	7	1	no	—
165	C₂²×C₄	—	7	2	no	—
166	C₂²×C₄	—	7	1	no	—
167	C₂²×C₄	—	7	2	no	—
168	C₂²×C₄	C₂²×D₇	1	2	yes	—
169	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
170	C₂²×C₄	C₂²×D₇	1	2	yes	—
171	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
172	C₂²×C₄	C₂²×D₇	1	1	yes	characteristic
173	C₄⋊C₄	—	2	2	no	—
174	C₄⋊C₄	—	7	8	no	—
175	C₄⋊C₄	—	7	4	no	—
176	C₄⋊C₄	—	14	4	no	—
177	C₄⋊C₄	—	14	4	no	—
178	C₄⋊C₄	—	14	4	no	—
179	C₄⋊C₄	—	14	4	no	—
180	C₄⋊C₄	—	14	2	no	—
181	C₄⋊C₄	C₄×D₇	1	4	yes	—
182	C₄²	—	2	4	no	—
183	C₄²	—	7	4	no	—
184	C₄²	—	7	4	no	—
185	C₄²	—	7	4	no	—
186	C₄²	—	14	2	no	—
187	C₄²	—	14	2	no	—
188	C₄²	—	14	2	no	—
189	C₄²	—	14	4	no	—
190	C₄²	Dic₁₄	1	4	yes	complemented
191	C₂×C₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
193	C₂×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
194	C₂×C₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
195	C₂×C₇	#532	1	1	yes	characteristic
196	C₂×C₇	#533	1	1	yes	characteristic
197	C₂×C₇	#533	1	1	yes	characteristic
198	C₂³	C₂³×D₇	1	1	yes	characteristic
199	Q₈	—	14	16	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	7	2	no	—
208	C₂×C₄	—	7	2	no	—
209	C₂×C₄	—	7	2	no	—
210	C₂×C₄	—	7	2	no	—
211	C₂×C₄	—	7	2	no	—
212	C₂×C₄	—	7	2	no	—
213	C₂×C₄	—	7	4	no	—
214	C₂×C₄	—	7	2	no	—
215	C₂×C₄	—	7	2	no	—
216	C₂×C₄	—	14	1	no	—
217	C₂×C₄	—	14	1	no	—
218	C₂×C₄	—	14	2	no	—
219	C₂×C₄	—	14	1	no	—
220	C₂×C₄	—	14	1	no	—
221	C₂×C₄	—	14	2	no	—
222	C₂×C₄	—	14	2	no	—
223	C₂×C₄	—	14	2	no	—
224	C₂×C₄	—	14	2	no	—
225	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
226	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
227	C₂×C₄	C₂×Dic₁₄	1	2	yes	—
228	C₂×C₄	C₂×C₄×D₇	1	4	yes	—
229	C₂×C₄	C₂×C₄×D₇	1	2	yes	—
230	C₂×C₄	#1157	1	2	yes	—
231	C₂×C₄	D₄×D₇	1	2	yes	—
232	C₂×C₄	Q₈×D₇	1	2	yes	—
233	C₇	C₄×C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
234	C₂²	C₂²×Dic₁₄	1	1	yes	characteristic
235	C₂²	C₂²×C₄×D₇	1	1	yes	characteristic
236	C₂²	C₂×#1157	1	1	yes	characteristic
237	C₂²	C₂×D₄×D₇	1	1	yes	characteristic
238	C₂²	C₂×Q₈×D₇	1	1	yes	characteristic
239	C₂²	D₇×#41	1	1	yes	characteristic
240	C₂²	D₇×#41	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	2	4	no	—
243	C₄	—	4	2	no	—
244	C₄	—	7	4	no	—
245	C₄	—	7	4	no	—
246	C₄	—	14	4	no	—
247	C₄	—	14	2	no	—
248	C₄	—	14	2	no	—
249	C₄	—	28	2	no	—
250	C₄	C₄×Dic₁₄	1	4	yes	complemented
251	C₄	#6451	1	4	yes	direct product factor
252	C₂	C₂×C₄×Dic₁₄	1	1	yes	characteristic
253	C₂	C₂×#6451	1	1	yes	characteristic
254	C₂	#90208	1	1	yes	characteristic
255	C₂	C₄×D₄×D₇	1	1	yes	characteristic
256	C₂	#90239	1	1	yes	characteristic
257	C₂	C₄×Q₈×D₇	1	1	yes	characteristic
258	C₂	#90269	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
31. #6451	C₄	direct (N × H)
55. C₄²×C₇	Q₈	semidirect
69. C₄×Dic₇	Q₈	semidirect
151. C₄×C₇	C₄×Q₈	semidirect
152. C₄×C₇	C₄⋊Q₈	semidirect
158. Dic₇	C₄×Q₈	semidirect
190. C₄²	Dic₁₄	semidirect
233. C₇	C₄×C₄⋊Q₈	semidirect
250. C₄	C₄×Dic₁₄	semidirect
251. C₄	#6451	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×#6451	2
C₂×C₄×Dic₁₄	2
C₄×C₇×C₄⋊C₄	2
#89757	2
#89748	2
#89745	2
C₄×C₄⋊Dic₇	2
C₄×Dic₇⋊C₄	2
C₄²×Dic₇	2
#89419	2
#89418	2
C₄×C₄⋊Q₈	7

Subgroups — C4×#6451 #11731069

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#6451 #11731069