Subgroups — C₄×#89348 #13268836

Order 1792.

522 conjugacy classes of subgroups.

The subgroup lattice has 264 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×#89348	C₂	1	2	yes	maximal
3	C₄×#6386	C₂	1	2	yes	maximal, complemented
4	#11725470	C₂	1	2	yes	maximal, complemented
5	C₂²×C₇×M₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₇×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₇	C₄	1	1	yes	characteristic
8	C₄×C₇×M₁₆	—	2	2	no	—
9	C₄×C₇×M₁₆	—	2	2	no	—
10	C₂×C₇×#97	C₄	1	1	yes	characteristic
11	C₂×C₇×#97	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₇×C₈	C₄	1	1	yes	characteristic
13	C₂×C₄×C₇×C₈	C₂²	1	1	yes	characteristic
14	C₂×#6386	C₂²	1	4	yes	—
15	#89348	C₄	1	8	yes	direct product factor
16	#89256	—	2	2	no	—
17	C₄×C₇⋊C₁₆	—	2	2	no	—
18	C₄×#349	—	7	1	no	maximal, Sylow
19	C₂³×C₄×C₇	C₈	1	2	yes	—
20	C₂³×C₄×C₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇×M₁₆	—	2	2	no	—
22	C₂×C₇×M₁₆	—	2	2	no	—
23	C₂×C₇×M₁₆	—	2	2	no	—
24	C₂×C₇×M₁₆	—	2	2	no	—
25	C₂×C₇×M₁₆	C₈	1	4	yes	—
26	C₂×C₇×M₁₆	C₂×C₄	1	4	yes	—
27	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₇×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₇×C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₇	—	2	1	no	—
32	C₂×C₄²×C₇	—	2	1	no	—
33	C₂×C₄²×C₇	—	2	2	no	—
34	C₂×C₄²×C₇	C₈	1	2	yes	—
35	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄²×C₇	C₂×C₄	1	1	yes	characteristic
37	C₇×#97	—	2	2	no	—
38	C₇×#97	—	2	2	no	—
39	C₄×C₇×C₈	—	2	1	no	—
40	C₄×C₇×C₈	—	2	1	no	—
41	C₄×C₇×C₈	D₄	1	2	yes	—
42	C₄×C₇×C₈	D₄	1	2	yes	—
43	#6386	—	2	4	no	—
44	#6386	C₂×C₄	1	8	yes	complemented
45	C₂×C₇⋊C₁₆	—	2	4	no	—
46	C₂×C₄×M₁₆	D₇	1	1	yes	characteristic
47	C₂×#349	—	7	2	no	—
48	C₄×M₃₂	—	7	2	no	—
49	#1377	—	7	2	no	—
50	C₂⁴×C₇	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₇	—	2	1	no	—
52	C₂²×C₄×C₇	—	2	1	no	—
53	C₂²×C₄×C₇	—	2	1	no	—
54	C₂²×C₄×C₇	—	2	1	no	—
55	C₂²×C₄×C₇	—	2	2	no	—
56	C₂²×C₄×C₇	—	2	2	no	—
57	C₂²×C₄×C₇	—	4	4	no	—
58	C₂²×C₄×C₇	C₄²	1	2	yes	—
59	C₂²×C₄×C₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
62	C₂²×C₄×C₇	C₂×C₈	1	2	yes	—
63	C₂²×C₄×C₇	C₂²×C₄	1	1	yes	characteristic
64	C₇×M₁₆	—	2	8	no	—
65	C₇×M₁₆	—	2	8	no	—
66	C₂×C₇×C₈	—	2	1	no	—
67	C₂×C₇×C₈	—	2	1	no	—
68	C₂×C₇×C₈	—	2	1	no	—
69	C₂×C₇×C₈	—	2	1	no	—
70	C₂×C₇×C₈	C₄²	1	2	yes	—
71	C₂×C₇×C₈	C₂²⋊C₄	1	2	yes	—
72	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
73	C₂×C₇×C₈	C₂×C₈	1	2	yes	—
74	C₂×C₇×C₈	M₁₆	1	2	yes	—
75	C₂×C₇×C₈	C₂²×C₄	1	2	yes	—
76	C₂×C₇×C₈	C₂×D₄	1	2	yes	—
77	C₂×C₇×C₈	#41	1	2	yes	—
78	C₄²×C₇	—	2	2	no	—
79	C₄²×C₇	—	2	2	no	—
80	C₄²×C₇	—	2	1	no	—
81	C₄²×C₇	—	2	1	no	—
82	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
83	C₄²×C₇	C₂²⋊C₄	1	1	yes	characteristic
84	C₄²×C₇	M₁₆	1	2	yes	—
85	C₇⋊C₁₆	—	2	8	no	—
86	C₂²×M₁₆	Dic₇	1	1	yes	characteristic
87	C₂²×M₁₆	C₂×D₇	1	1	yes	characteristic
88	C₂²×C₄²	Dic₇	1	1	yes	characteristic
89	C₂×M₃₂	—	7	4	no	—
90	C₄×M₁₆	—	2	2	no	—
91	C₄×M₁₆	—	2	2	no	—
92	C₂×#97	Dic₇	1	1	yes	characteristic
93	C₂×#97	C₂×D₇	1	1	yes	characteristic
94	C₂×C₄×C₈	Dic₇	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₇	1	1	yes	characteristic
96	#349	—	7	8	no	—
97	#346	—	14	2	no	—
98	C₄×C₁₆	—	14	2	no	—
99	C₂³×C₇	—	2	1	no	—
100	C₂³×C₇	—	2	1	no	—
101	C₂³×C₇	—	4	2	no	—
102	C₂³×C₇	C₄×C₈	1	2	yes	—
103	C₂³×C₇	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₇	—	2	1	no	—
105	C₂×C₄×C₇	—	2	1	no	—
106	C₂×C₄×C₇	—	2	4	no	—
107	C₂×C₄×C₇	—	2	2	no	—
108	C₂×C₄×C₇	—	2	2	no	—
109	C₂×C₄×C₇	—	2	2	no	—
110	C₂×C₄×C₇	—	2	2	no	—
111	C₂×C₄×C₇	—	2	2	no	—
112	C₂×C₄×C₇	—	4	4	no	—
113	C₂×C₄×C₇	—	4	4	no	—
114	C₂×C₄×C₇	—	4	2	no	—
115	C₂×C₄×C₇	C₄×C₈	1	2	yes	—
116	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
117	C₂×C₄×C₇	C₂²⋊C₈	1	2	yes	—
118	C₂×C₄×C₇	C₂×C₄²	1	1	yes	characteristic
119	C₂×C₄×C₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	#117	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
123	C₂×C₄×C₇	C₂×M₁₆	1	1	yes	characteristic
124	C₇×C₈	—	2	2	no	—
125	C₇×C₈	—	2	2	no	—
126	C₇×C₈	C₂²⋊C₈	1	4	yes	—
127	C₇×C₈	C₄×D₄	1	4	yes	—
128	C₂³×C₄	C₇⋊C₈	1	2	yes	—
129	C₂³×C₄	C₂×Dic₇	1	1	yes	characteristic
130	C₂×M₁₆	—	2	2	no	—
131	C₂×M₁₆	—	2	2	no	—
132	C₂×M₁₆	—	2	2	no	—
133	C₂×M₁₆	—	2	2	no	—
134	C₂×M₁₆	C₇⋊C₈	1	4	yes	—
135	C₂×M₁₆	C₄×D₇	1	4	yes	—
136	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
137	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
138	C₂²×C₈	C₂×Dic₇	1	1	yes	characteristic
139	C₂²×C₈	C₂²×D₇	1	1	yes	characteristic
140	C₂×C₄²	—	2	2	no	—
141	C₂×C₄²	—	2	1	no	—
142	C₂×C₄²	—	2	1	no	—
143	C₂×C₄²	C₇⋊C₈	1	2	yes	—
144	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
145	C₂×C₄²	C₂×Dic₇	1	1	yes	characteristic
146	M₃₂	—	7	8	no	—
147	M₃₂	—	14	4	no	—
148	C₂×C₁₆	—	14	4	no	—
149	#97	—	2	2	no	—
150	#97	—	2	2	no	—
151	C₄×C₈	—	2	1	no	—
152	C₄×C₈	—	2	1	no	—
153	C₄×C₈	C₂²⋊D₇	1	2	yes	—
154	C₄×C₈	C₂²⋊D₇	1	2	yes	—
155	C₂²×C₇	—	2	2	no	—
156	C₂²×C₇	—	2	2	no	—
157	C₂²×C₇	—	4	2	no	—
158	C₂²×C₇	—	4	2	no	—
159	C₂²×C₇	—	4	2	no	—
160	C₂²×C₇	C₂×C₄×C₈	1	1	yes	characteristic
161	C₂²×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
162	C₂²×C₇	#431	1	1	yes	characteristic
163	C₄×C₇	—	2	2	no	—
164	C₄×C₇	—	2	2	no	—
165	C₄×C₇	—	4	2	no	—
166	C₄×C₇	#349	1	4	yes	complemented
167	C₄×C₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
168	C₄×C₇	C₄×M₁₆	1	1	yes	characteristic
169	C₄×C₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₇	#431	1	1	yes	characteristic
171	C₂⁴	C₂×C₇⋊C₈	1	1	yes	characteristic
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	2	1	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	4	4	no	—
179	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
180	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
181	C₂²×C₄	C₂×C₇⋊C₈	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₇⋊C₈	1	2	yes	—
183	C₂²×C₄	C₄×Dic₇	1	2	yes	—
184	C₂²×C₄	C₂²×Dic₇	1	1	yes	characteristic
185	M₁₆	—	2	8	no	—
186	M₁₆	—	2	8	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	2	1	no	—
190	C₂×C₈	—	2	1	no	—
191	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
192	C₂×C₈	C₂×C₇⋊C₈	1	2	yes	—
193	C₂×C₈	#1136	1	2	yes	—
194	C₂×C₈	C₄×Dic₇	1	2	yes	—
195	C₂×C₈	C₂²⋊Dic₇	1	2	yes	—
196	C₂×C₈	C₂×C₄×D₇	1	2	yes	—
197	C₂×C₈	#1157	1	2	yes	—
198	C₂×C₈	C₂×C₂²⋊D₇	1	2	yes	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	#1136	1	2	yes	—
204	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
205	C₄²	C₂²⋊Dic₇	1	1	yes	characteristic
206	C₁₆	—	14	8	no	—
207	C₂×C₇	—	2	1	no	—
208	C₂×C₇	—	2	1	no	—
209	C₂×C₇	—	4	2	no	—
210	C₂×C₇	C₄×C₂²⋊C₈	1	1	yes	characteristic
211	C₂×C₇	C₂×#349	1	2	yes	—
212	C₂³	—	2	1	no	—
213	C₂³	—	2	1	no	—
214	C₂³	—	4	2	no	—
215	C₂³	C₄×C₇⋊C₈	1	2	yes	—
216	C₂³	C₂²×C₇⋊C₈	1	1	yes	characteristic
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₄×C₇⋊C₈	1	2	yes	—
229	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
230	C₂×C₄	C₂²⋊(C₇⋊C₈)	1	2	yes	—
231	C₂×C₄	C₂²×C₇⋊C₈	1	1	yes	characteristic
232	C₂×C₄	C₂×#1136	1	1	yes	characteristic
233	C₂×C₄	C₂×#1136	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×Dic₇	1	1	yes	characteristic
235	C₂×C₄	#6489	1	1	yes	characteristic
236	C₂×C₄	C₂×C₂²⋊Dic₇	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	—	2	2	no	—
239	C₈	C₂²⋊(C₇⋊C₈)	1	4	yes	—
240	C₈	C₄×C₂²⋊D₇	1	4	yes	—
241	C₇	C₄×#349	1	1	yes	characteristic, complemented, Sylow
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	2	no	—
245	C₂²	—	4	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	C₂×C₄×C₇⋊C₈	1	1	yes	characteristic
248	C₂²	#89697	1	1	yes	characteristic
249	C₂²	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	2	no	—
252	C₄	—	4	2	no	—
253	C₄	#89348	1	4	yes	direct product factor
254	C₄	C₄×#1136	1	1	yes	characteristic
255	C₄	#89697	1	1	yes	characteristic
256	C₄	C₂×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
257	C₄	C₄×C₂²⋊Dic₇	1	1	yes	characteristic
258	C₂	—	2	1	no	—
259	C₂	—	2	1	no	—
260	C₂	—	4	2	no	—
261	C₂	C₄×C₂²⋊(C₇⋊C₈)	1	1	yes	characteristic
262	C₂	C₂×#89348	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²⋊Dic₇
commutator	C₂²×C₇	C₂×C₄×C₈
Frattini	C₂²×C₈	C₂²×D₇
Fitting	C₂×C₄×C₇×M₁₆	C₂
socle	C₂²×C₇	C₂×C₂²⋊C₈
prime radical	C₂⁴×C₇	C₂×C₈
cosocle	C₂²×C₇×C₈	C₂³
hypercenter	C₂×C₄×M₁₆	D₇
hypocenter	C₇	C₄×#349
LCS	G ▷ C₂²×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. C₄×#6386	C₂	semidirect
4. #11725470	C₂	semidirect
15. #89348	C₄	direct (N × H)
44. #6386	C₂×C₄	semidirect
166. C₄×C₇	#349	semidirect
241. C₇	C₄×#349	semidirect
253. C₄	#89348	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×#89348	2
C₄×#6386	2
#11725470	2
C₄×#349	7

Subgroups — C4×#89348 #13268836

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#89348 #13268836