Subgroups — C₇×Q₈×C₄⋊C₈ #13199936

Order 1792.

896 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₇×C₈×Q₈	C₂	1	2	yes	maximal
2	C₄²×C₇×Q₈	C₂	1	1	yes	maximal, characteristic
3	C₇×#1939	C₂	1	6	yes	maximal
4	C₇×#1842	C₂	1	3	yes	maximal
5	C₄×C₇×C₄⋊C₈	C₂	1	3	yes	maximal
6	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄×C₇×Q₈	C₄	1	1	yes	characteristic
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×C₈×Q₈	—	2	8	no	—
10	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
11	C₂×C₇×C₄⋊C₈	C₂²	1	3	yes	—
12	C₂×C₇×C₄⋊C₈	C₂²	1	6	yes	—
13	C₂×C₇×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₇×C₈	C₂²	1	6	yes	—
15	C₄×C₇×C₄⋊C₄	C₄	1	3	yes	—
16	C₄×C₇×C₄⋊C₄	C₄	1	3	yes	—
17	C₄×C₇×C₄⋊C₄	C₂²	1	3	yes	—
18	C₄³×C₇	C₂²	1	3	yes	—
19	C₇×#336	C₂²	1	6	yes	—
20	Q₈×C₄⋊C₈	C₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₇×C₈	C₂³	1	6	yes	—
23	C₂²×C₇×C₈	C₂³	1	2	yes	—
24	C₄×C₇×Q₈	—	2	4	no	—
25	C₄×C₇×Q₈	C₈	1	8	yes	complemented
26	C₄×C₇×Q₈	D₄	1	4	yes	—
27	C₄×C₇×Q₈	Q₈	1	4	yes	—
28	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
29	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
30	C₂×C₇×C₄⋊C₄	C₂×C₄	1	3	yes	—
31	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
32	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
33	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
34	C₂×C₄²×C₇	C₂×C₄	1	3	yes	—
35	C₂×C₄²×C₇	C₂³	1	3	yes	—
36	C₂×C₄²×C₇	C₂³	1	3	yes	—
37	C₂×C₄²×C₇	C₂³	1	1	yes	characteristic
38	C₇×C₄⋊C₈	—	2	12	no	—
39	C₇×C₄⋊C₈	—	2	6	no	—
40	C₇×C₄⋊C₈	—	4	6	no	—
41	C₇×C₄⋊C₈	Q₈	1	4	yes	direct product factor
42	C₄×C₇×C₈	—	2	12	no	—
43	C₇×#95	C₂×C₄	1	6	yes	—
44	C₂×C₈×Q₈	C₂×C₇	1	2	yes	—
45	C₄²×Q₈	C₂×C₇	1	1	yes	characteristic
46	#1939	C₂×C₇	1	6	yes	—
47	#1842	C₂×C₇	1	3	yes	—
48	C₄×C₄⋊C₈	C₂×C₇	1	3	yes	—
49	C₂×C₇×Q₈	C₄⋊C₄	1	4	yes	—
50	C₂×C₇×Q₈	C₂×C₈	1	4	yes	—
51	C₂×C₇×Q₈	M₁₆	1	4	yes	—
52	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
53	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
54	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
56	C₂²×C₄×C₇	C₂²×C₄	1	3	yes	—
57	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
59	C₂×C₇×C₈	—	2	2	no	—
60	C₂×C₇×C₈	—	2	2	no	—
61	C₂×C₇×C₈	—	2	6	no	—
62	C₂×C₇×C₈	—	2	6	no	—
63	C₂×C₇×C₈	—	2	6	no	—
64	C₂×C₇×C₈	C₂×Q₈	1	4	yes	—
65	C₇×C₄⋊C₄	—	2	6	no	—
66	C₇×C₄⋊C₄	C₄⋊C₄	1	12	yes	—
67	C₇×C₄⋊C₄	C₂×C₈	1	12	yes	—
68	C₄²×C₇	—	2	6	no	—
69	C₄²×C₇	—	2	6	no	—
70	C₄²×C₇	—	2	3	no	—
71	C₄²×C₇	—	2	3	no	—
72	C₄²×C₇	—	4	3	no	—
73	C₄²×C₇	C₂×C₈	1	12	yes	—
74	C₄²×C₇	C₂×D₄	1	6	yes	—
75	C₄²×C₇	C₂×Q₈	1	6	yes	—
76	C₄²×C₇	C₂×Q₈	1	2	yes	—
77	C₄²×C₇	#41	1	2	yes	—
78	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
79	C₂×C₄×Q₈	C₄×C₇	1	1	yes	characteristic
80	C₂×C₄×Q₈	C₂²×C₇	1	1	yes	characteristic
81	C₈×Q₈	—	2	8	no	—
82	C₂×C₄⋊C₈	C₂²×C₇	1	3	yes	—
83	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
84	C₂×C₄⋊C₈	C₂²×C₇	1	6	yes	—
85	C₂×C₄⋊C₈	C₂²×C₇	1	1	yes	characteristic
86	C₂×C₄×C₈	C₂²×C₇	1	6	yes	—
87	C₄×C₄⋊C₄	C₄×C₇	1	3	yes	—
88	C₄×C₄⋊C₄	C₄×C₇	1	3	yes	—
89	C₄×C₄⋊C₄	C₂²×C₇	1	3	yes	—
90	C₄³	C₂²×C₇	1	3	yes	—
91	#336	C₂²×C₇	1	6	yes	—
92	C₂³×C₇	C₂³×C₄	1	1	yes	characteristic
93	C₇×Q₈	C₄⋊C₈	1	16	yes	direct product factor
94	C₂×C₄×C₇	—	2	3	no	—
95	C₂×C₄×C₇	—	2	1	no	—
96	C₂×C₄×C₇	—	2	3	no	—
97	C₂×C₄×C₇	—	2	1	no	—
98	C₂×C₄×C₇	—	2	3	no	—
99	C₂×C₄×C₇	—	2	3	no	—
100	C₂×C₄×C₇	—	2	3	no	—
101	C₂×C₄×C₇	—	2	3	no	—
102	C₂×C₄×C₇	—	2	3	no	—
103	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
104	C₂×C₄×C₇	C₂×C₄⋊C₄	1	6	yes	—
105	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
106	C₂×C₄×C₇	C₄×Q₈	1	2	yes	—
107	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
108	C₂×C₄×C₇	C₂²×C₈	1	6	yes	—
109	C₂×C₄×C₇	C₂²×C₈	1	2	yes	—
110	C₂×C₄×C₇	C₂×M₁₆	1	6	yes	—
111	C₂×C₄×C₇	#131	1	2	yes	—
112	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₇	C₂²×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
116	C₂×C₄×C₇	H(5,𝔽₂)	1	1	yes	characteristic
117	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₇×C₈	—	2	4	no	—
119	C₇×C₈	—	4	6	no	—
120	C₂²×Q₈	C₂×C₄×C₇	1	1	yes	characteristic
121	C₂²×C₈	C₂³×C₇	1	6	yes	—
122	C₂²×C₈	C₂³×C₇	1	2	yes	—
123	C₄×Q₈	—	2	4	no	—
124	C₄×Q₈	C₇×C₈	1	8	yes	complemented
125	C₄×Q₈	C₇×D₄	1	4	yes	—
126	C₄×Q₈	C₇×Q₈	1	4	yes	—
127	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
128	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
129	C₂×C₄⋊C₄	C₂×C₄×C₇	1	3	yes	—
130	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
131	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
132	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
133	C₂×C₄²	C₂×C₄×C₇	1	3	yes	—
134	C₂×C₄²	C₂³×C₇	1	3	yes	—
135	C₂×C₄²	C₂³×C₇	1	3	yes	—
136	C₂×C₄²	C₂³×C₇	1	1	yes	characteristic
137	C₄⋊C₈	—	2	12	no	—
138	C₄⋊C₈	—	2	6	no	—
139	C₄⋊C₈	—	4	6	no	—
140	C₄⋊C₈	C₇×Q₈	1	4	yes	direct product factor
141	C₄×C₈	—	2	12	no	—
142	#95	C₂×C₄×C₇	1	6	yes	—
143	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₂²×C₇	#520	1	1	yes	characteristic
146	C₂²×C₇	C₂³×C₈	1	1	yes	characteristic
147	C₂²×C₇	C₂²×M₁₆	1	1	yes	characteristic
148	C₂²×C₇	C₂×#131	1	1	yes	characteristic
149	C₂²×C₇	#568	1	1	yes	characteristic
150	C₄×C₇	—	2	6	no	—
151	C₄×C₇	—	2	2	no	—
152	C₄×C₇	—	2	6	no	—
153	C₄×C₇	—	4	3	no	—
154	C₄×C₇	C₂×C₄⋊C₈	1	12	yes	—
155	C₄×C₇	C₈×Q₈	1	4	yes	complemented
156	C₄×C₇	D₄×Q₈	1	1	yes	characteristic
157	C₄×C₇	#550	1	1	yes	characteristic
158	C₄×C₇	#557	1	1	yes	characteristic
159	C₄×C₇	Q₈²	1	1	yes	characteristic
160	C₂×Q₈	C₇×C₄⋊C₄	1	4	yes	—
161	C₂×Q₈	C₂×C₇×C₈	1	4	yes	—
162	C₂×Q₈	C₇×M₁₆	1	4	yes	—
163	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
164	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
165	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
166	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
167	C₂²×C₄	C₂²×C₄×C₇	1	3	yes	—
168	C₂²×C₄	C₂²×C₄×C₇	1	1	yes	characteristic
169	C₂²×C₄	C₂⁴×C₇	1	1	yes	characteristic
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	2	2	no	—
172	C₂×C₈	—	2	6	no	—
173	C₂×C₈	—	2	6	no	—
174	C₂×C₈	—	2	6	no	—
175	C₂×C₈	C₂×C₇×Q₈	1	4	yes	—
176	C₄⋊C₄	—	2	6	no	—
177	C₄⋊C₄	C₇×C₄⋊C₄	1	12	yes	—
178	C₄⋊C₄	C₂×C₇×C₈	1	12	yes	—
179	C₄²	—	2	6	no	—
180	C₄²	—	2	6	no	—
181	C₄²	—	2	3	no	—
182	C₄²	—	2	3	no	—
183	C₄²	—	4	3	no	—
184	C₄²	C₂×C₇×C₈	1	12	yes	—
185	C₄²	C₂×C₇×D₄	1	6	yes	—
186	C₄²	C₂×C₇×Q₈	1	6	yes	—
187	C₄²	C₂×C₇×Q₈	1	2	yes	—
188	C₄²	C₇×#41	1	2	yes	—
189	C₂×C₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
190	C₂×C₇	C₂²×C₄⋊C₈	1	1	yes	characteristic
191	C₂×C₇	#2906	1	1	yes	characteristic
192	C₂×C₇	C₂×C₈×Q₈	1	1	yes	characteristic
193	C₂×C₇	Q₈×M₁₆	1	1	yes	characteristic
194	C₂×C₇	#2988	1	1	yes	characteristic
195	C₂×C₇	#2991	1	1	yes	characteristic
196	C₂³	C₂³×C₄×C₇	1	1	yes	characteristic
197	Q₈	C₇×C₄⋊C₈	1	16	yes	direct product factor
198	C₂×C₄	—	2	3	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	3	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	3	no	—
203	C₂×C₄	—	2	3	no	—
204	C₂×C₄	—	2	3	no	—
205	C₂×C₄	—	2	3	no	—
206	C₂×C₄	—	2	3	no	—
207	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
208	C₂×C₄	C₂×C₇×C₄⋊C₄	1	6	yes	—
209	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
210	C₂×C₄	C₄×C₇×Q₈	1	2	yes	—
211	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
212	C₂×C₄	C₂²×C₇×C₈	1	6	yes	—
213	C₂×C₄	C₂²×C₇×C₈	1	2	yes	—
214	C₂×C₄	C₂×C₇×M₁₆	1	6	yes	—
215	C₂×C₄	C₇×#131	1	2	yes	—
216	C₂×C₄	C₂²×C₇×D₄	1	1	yes	characteristic
217	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₇×Q₈	1	1	yes	characteristic
219	C₂×C₄	C₂×C₇×#41	1	1	yes	characteristic
220	C₂×C₄	C₇×H(5,𝔽₂)	1	1	yes	characteristic
221	C₂×C₄	C₇×2₋¹⁺⁴	1	1	yes	characteristic
222	C₈	—	2	4	no	—
223	C₈	—	4	6	no	—
224	C₇	Q₈×C₄⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
225	C₂²	C₂²×C₇×C₄⋊C₄	1	1	yes	characteristic
226	C₂²	C₂×C₄×C₇×Q₈	1	1	yes	characteristic
227	C₂²	C₇×#520	1	1	yes	characteristic
228	C₂²	C₂³×C₇×C₈	1	1	yes	characteristic
229	C₂²	C₂²×C₇×M₁₆	1	1	yes	characteristic
230	C₂²	C₂×C₇×#131	1	1	yes	characteristic
231	C₂²	C₇×#568	1	1	yes	characteristic
232	C₄	—	2	6	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	6	no	—
235	C₄	—	4	3	no	—
236	C₄	C₂×C₇×C₄⋊C₈	1	12	yes	—
237	C₄	C₇×C₈×Q₈	1	4	yes	complemented
238	C₄	C₇×D₄×Q₈	1	1	yes	characteristic
239	C₄	C₇×#550	1	1	yes	characteristic
240	C₄	C₇×#557	1	1	yes	characteristic
241	C₄	C₇×Q₈²	1	1	yes	characteristic
242	C₂	C₇×Q₈×C₄⋊C₄	1	1	yes	characteristic
243	C₂	C₂²×C₇×C₄⋊C₈	1	1	yes	characteristic
244	C₂	C₇×#2906	1	1	yes	characteristic
245	C₂	C₂×C₇×C₈×Q₈	1	1	yes	characteristic
246	C₂	C₇×Q₈×M₁₆	1	1	yes	characteristic
247	C₂	C₇×#2988	1	1	yes	characteristic
248	C₂	C₇×#2991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₇	C₂⁴
commutator	C₂²	C₂³×C₇×C₈
Frattini, cosocle	C₂²×C₄	C₂⁴×C₇
socle, prime radical	C₂³×C₇	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₇ ◁ G

Complements

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

N	H ≅ G/N	type
20. Q₈×C₄⋊C₈	C₇	direct (N × H)
25. C₄×C₇×Q₈	C₈	semidirect
41. C₇×C₄⋊C₈	Q₈	direct (N × H)
93. C₇×Q₈	C₄⋊C₈	direct (N × H)
124. C₄×Q₈	C₇×C₈	semidirect
140. C₄⋊C₈	C₇×Q₈	direct (N × H)
155. C₄×C₇	C₈×Q₈	semidirect
197. Q₈	C₇×C₄⋊C₈	direct (N × H)
224. C₇	Q₈×C₄⋊C₈	direct (N × H)
237. C₄	C₇×C₈×Q₈	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₇×C₈×Q₈	2
C₄²×C₇×Q₈	2
C₇×#1939	2
C₇×#1842	2
C₄×C₇×C₄⋊C₈	2
Q₈×C₄⋊C₈	7

Subgroups — C7×Q8×C4⋊C8 #13199936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₇×Q₈×C₄⋊C₈ #13199936