Subgroups — #13372798

Order 1792.

642 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₇×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic
2	Q₈×#1136	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈×C₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#11728274	C₂	1	3	yes	maximal
5	#11728270	C₂	1	3	yes	maximal
6	#11727629	C₂	1	3	yes	maximal
7	C₄×#6396	C₂	1	3	yes	maximal
8	C₂×C₄×C₇×Q₈	C₂²	1	1	yes	characteristic
9	C₇×#446	—	2	3	no	—
10	C₇×C₈×Q₈	—	2	1	no	—
11	C₇×#423	C₂²	1	3	yes	—
12	C₄×C₇×M₁₆	C₂²	1	3	yes	—
13	C₂×#6396	C₂²	1	3	yes	—
14	C₂×#6396	C₂²	1	3	yes	—
15	#89796	—	2	3	no	—
16	Q₈×C₇⋊C₈	—	2	1	no	—
17	Q₈×C₇⋊C₈	C₄	1	6	yes	—
18	Q₈×C₇⋊C₈	C₄	1	2	yes	—
19	#89696	C₂²	1	3	yes	—
20	C₂×C₄⋊(C₇⋊C₈)	C₂²	1	3	yes	—
21	C₄×#1136	C₂²	1	3	yes	—
22	C₂×C₄×C₇⋊C₈	C₂²	1	3	yes	—
23	#89353	C₂²	1	1	yes	characteristic
24	#89353	C₂²	1	3	yes	—
25	#89353	C₂²	1	6	yes	—
26	#22746	—	7	1	no	maximal, Sylow
27	C₂²×C₇×Q₈	Q₈	1	1	yes	characteristic
28	C₂×C₇×M₁₆	C₂³	1	3	yes	—
29	C₂×C₇×M₁₆	C₂³	1	1	yes	characteristic
30	C₄×C₇×Q₈	C₂×C₄	1	3	yes	—
31	C₄×C₇×Q₈	C₂×C₄	1	1	yes	characteristic
32	C₄×C₇×Q₈	D₄	1	1	yes	characteristic
33	C₄×C₇×Q₈	D₄	1	3	yes	—
34	C₂×C₇×C₄⋊C₄	Q₈	1	3	yes	—
35	C₂×C₄²×C₇	C₂³	1	3	yes	—
36	C₂×#1136	C₂³	1	3	yes	—
37	C₂×#1136	C₂³	1	1	yes	characteristic
38	C₂²×C₇⋊C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₇⋊C₈	C₂³	1	3	yes	—
40	C₇×C₄⋊C₈	—	2	3	no	—
41	C₇×C₄⋊C₈	—	2	3	no	—
42	C₇×#97	—	2	3	no	—
43	C₄×C₇×C₈	—	2	3	no	—
44	#6396	—	2	6	no	—
45	#6396	—	2	6	no	—
46	C₄⋊(C₇⋊C₈)	—	2	3	no	—
47	C₄⋊(C₇⋊C₈)	—	2	3	no	—
48	C₄⋊(C₇⋊C₈)	C₂×C₄	1	6	yes	—
49	C₄⋊(C₇⋊C₈)	C₂×C₄	1	6	yes	—
50	#6377	—	2	3	no	—
51	C₄×C₇⋊C₈	—	2	3	no	—
52	C₄×C₇⋊C₈	C₂×C₄	1	6	yes	—
53	C₄×C₇⋊C₈	C₂×C₄	1	6	yes	—
54	Q₈×M₁₆	—	7	1	no	—
55	Q₈×M₁₆	D₇	1	1	yes	characteristic
56	C₂×C₈×Q₈	—	7	1	no	—
57	#1950	—	7	3	no	—
58	#1950	—	7	3	no	—
59	#1837	—	7	3	no	—
60	C₄×#108	—	7	3	no	—
61	C₂×C₇×Q₈	—	2	4	no	—
62	C₂×C₇×Q₈	C₄⋊C₄	1	3	yes	—
63	C₂×C₇×Q₈	C₄⋊C₄	1	1	yes	characteristic
64	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
65	C₂²×C₄×C₇	C₂×Q₈	1	3	yes	—
66	C₂²×C₄×C₇	C₂⁴	1	1	yes	characteristic
67	C₇×M₁₆	—	2	1	no	—
68	C₇×M₁₆	—	2	3	no	—
69	C₇×M₁₆	—	2	3	no	—
70	C₇×M₁₆	#41	1	2	yes	—
71	C₂×C₇×C₈	—	2	1	no	—
72	C₂×C₇×C₈	—	2	3	no	—
73	C₇×C₄⋊C₄	C₄⋊C₄	1	6	yes	—
74	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
75	C₇×C₄⋊C₄	C₄⋊C₄	1	3	yes	—
76	C₄²×C₇	C₂²×C₄	1	3	yes	—
77	C₄²×C₇	C₂²×C₄	1	3	yes	—
78	C₄²×C₇	C₂×D₄	1	3	yes	—
79	C₄²×C₇	C₂×D₄	1	3	yes	—
80	#1136	—	2	1	no	—
81	#1136	—	2	3	no	—
82	#1136	—	2	3	no	—
83	#1136	#41	1	2	yes	—
84	C₂×C₇⋊C₈	—	2	1	no	—
85	C₂×C₇⋊C₈	—	2	3	no	—
86	C₂×C₇⋊C₈	—	2	3	no	—
87	C₂×C₇⋊C₈	—	2	1	no	—
88	C₂×C₇⋊C₈	—	2	1	no	—
89	C₂×C₇⋊C₈	—	2	3	no	—
90	C₂×C₇⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₇⋊C₈	C₂²×C₄	1	6	yes	—
92	C₂×C₄×Q₈	C₂×D₇	1	1	yes	characteristic
93	#446	—	2	3	no	—
94	#446	—	14	3	no	—
95	C₈×Q₈	—	2	1	no	—
96	C₈×Q₈	—	7	6	no	—
97	C₈×Q₈	—	7	2	no	—
98	C₈×Q₈	—	14	1	no	—
99	C₂×#108	—	7	3	no	—
100	C₂×#108	—	7	3	no	—
101	#423	—	7	3	no	—
102	#423	C₂×D₇	1	3	yes	—
103	C₂×C₄⋊C₈	—	7	3	no	—
104	C₄×M₁₆	—	7	3	no	—
105	C₄×M₁₆	C₂×D₇	1	3	yes	—
106	C₂×C₄×C₈	—	7	3	no	—
107	#341	—	7	3	no	—
108	#341	—	7	6	no	—
109	#341	—	7	1	no	—
110	C₂³×C₇	C₂²×Q₈	1	1	yes	characteristic
111	C₇×Q₈	—	2	6	no	—
112	C₇×Q₈	#108	1	4	yes	complemented
113	C₂×C₄×C₇	—	2	6	no	—
114	C₂×C₄×C₇	—	4	3	no	—
115	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
116	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
117	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
118	C₂×C₄×C₇	C₂×C₄⋊C₄	1	3	yes	—
119	C₂×C₄×C₇	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
122	C₂×C₄×C₇	C₂×#41	1	1	yes	characteristic
123	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
124	C₂×C₄×C₇	2₋¹⁺⁴	1	1	yes	characteristic
125	C₇×C₈	—	2	2	no	—
126	C₇×C₈	—	4	3	no	—
127	C₇⋊C₈	—	2	6	no	—
128	C₇⋊C₈	—	2	2	no	—
129	C₇⋊C₈	—	2	2	no	—
130	C₇⋊C₈	—	4	3	no	—
131	C₂²×Q₈	Dic₁₄	1	1	yes	characteristic
132	C₂×M₁₆	—	7	1	no	—
133	C₂×M₁₆	—	7	3	no	—
134	C₂×M₁₆	C₂²×D₇	1	3	yes	—
135	C₂×M₁₆	C₂²×D₇	1	1	yes	characteristic
136	C₂²×C₈	—	7	1	no	—
137	C₂²×C₈	—	7	3	no	—
138	C₄×Q₈	C₄×D₇	1	3	yes	—
139	C₄×Q₈	C₄×D₇	1	1	yes	characteristic
140	C₄×Q₈	C₂²⋊D₇	1	1	yes	characteristic
141	C₄×Q₈	C₂²⋊D₇	1	3	yes	—
142	C₂×C₄⋊C₄	Dic₁₄	1	3	yes	—
143	C₂×C₄²	C₂²×D₇	1	3	yes	—
144	#108	—	14	6	no	—
145	#108	—	14	6	no	—
146	C₄⋊C₈	—	2	3	no	—
147	C₄⋊C₈	—	2	3	no	—
148	C₄⋊C₈	—	7	6	no	—
149	C₄⋊C₈	—	7	6	no	—
150	C₄⋊C₈	—	14	3	no	—
151	C₄⋊C₈	—	14	3	no	—
152	#97	—	2	3	no	—
153	#97	—	14	3	no	—
154	C₄×C₈	—	2	3	no	—
155	C₄×C₈	—	7	6	no	—
156	C₄×C₈	—	7	6	no	—
157	C₄×C₈	—	14	3	no	—
158	C₂²×C₇	—	2	1	no	—
159	C₂²×C₇	—	2	1	no	—
160	C₂²×C₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
161	C₂²×C₇	#557	1	1	yes	characteristic
162	C₂²×C₇	#557	1	1	yes	characteristic
163	C₄×C₇	—	2	3	no	—
164	C₄×C₇	—	2	3	no	—
165	C₄×C₇	—	2	3	no	—
166	C₄×C₇	C₂×#108	1	6	yes	—
167	C₄×C₇	C₄×#41	1	1	yes	characteristic
168	C₄×C₇	#519	1	1	yes	characteristic
169	C₄×C₇	#548	1	1	yes	characteristic
170	C₄×C₇	#548	1	1	yes	characteristic
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	Dic₇⋊C₄	1	3	yes	—
173	C₂×Q₈	Dic₇⋊C₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
175	C₂²×C₄	C₂×Dic₁₄	1	3	yes	—
176	C₂²×C₄	C₂³×D₇	1	1	yes	characteristic
177	M₁₆	—	2	1	no	—
178	M₁₆	—	2	3	no	—
179	M₁₆	—	2	3	no	—
180	M₁₆	—	7	2	no	—
181	M₁₆	—	14	1	no	—
182	M₁₆	—	14	3	no	—
183	M₁₆	—	14	3	no	—
184	M₁₆	#1161	1	2	yes	—
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	3	no	—
187	C₂×C₈	—	7	2	no	—
188	C₂×C₈	—	7	6	no	—
189	C₂×C₈	—	14	3	no	—
190	C₂×C₈	—	14	1	no	—
191	C₂×C₈	—	14	3	no	—
192	C₂×C₈	—	14	1	no	—
193	C₂×C₈	—	14	1	no	—
194	C₂×C₈	—	14	3	no	—
195	C₄⋊C₄	Dic₇⋊C₄	1	6	yes	—
196	C₄⋊C₄	Dic₇⋊C₄	1	3	yes	—
197	C₄⋊C₄	Dic₇⋊C₄	1	3	yes	—
198	C₄²	C₂×C₄×D₇	1	3	yes	—
199	C₄²	C₂×C₄×D₇	1	3	yes	—
200	C₄²	C₂×C₂²⋊D₇	1	3	yes	—
201	C₄²	C₂×C₂²⋊D₇	1	3	yes	—
202	C₂×C₇	—	2	1	no	—
203	C₂×C₇	—	2	1	no	—
204	C₂×C₇	#2351	1	1	yes	characteristic
205	C₂×C₇	C₂²×#108	1	1	yes	characteristic
206	C₂×C₇	#2916	1	1	yes	characteristic
207	C₂³	C₂²×Dic₁₄	1	1	yes	characteristic
208	Q₈	—	2	6	no	—
209	Q₈	#6396	1	4	yes	complemented
210	C₂×C₄	—	2	6	no	—
211	C₂×C₄	—	4	3	no	—
212	C₂×C₄	C₂×Dic₇⋊C₄	1	3	yes	—
213	C₂×C₄	C₂×Dic₇⋊C₄	1	3	yes	—
214	C₂×C₄	C₂×Dic₇⋊C₄	1	3	yes	—
215	C₂×C₄	C₂×Dic₇⋊C₄	1	3	yes	—
216	C₂×C₄	C₂²×C₄×D₇	1	1	yes	characteristic
217	C₂×C₄	C₂×#1161	1	1	yes	characteristic
218	C₂×C₄	#6551	1	1	yes	characteristic
219	C₂×C₄	D₇×#41	1	1	yes	characteristic
220	C₂×C₄	#6554	1	1	yes	characteristic
221	C₂×C₄	C₂²×C₂²⋊D₇	1	1	yes	characteristic
222	C₈	—	2	2	no	—
223	C₈	—	4	3	no	—
224	C₈	—	14	2	no	—
225	C₈	—	14	6	no	—
226	C₈	—	14	2	no	—
227	C₈	—	28	3	no	—
228	C₇	#22746	1	1	yes	characteristic, complemented, Sylow
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#90260	1	1	yes	characteristic
232	C₂²	#90261	1	1	yes	characteristic
233	C₂²	C₂²×Dic₇⋊C₄	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#6396	1	6	yes	—
238	C₄	#90263	1	1	yes	characteristic
239	C₄	C₄×#1161	1	1	yes	characteristic
240	C₄	#90508	1	1	yes	characteristic
241	C₄	#90522	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#11739906	1	1	yes	characteristic
245	C₂	C₂²×#6396	1	1	yes	characteristic
246	C₂	#11741255	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇
commutator	C₂×C₄×C₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇
Fitting	C₇×Q₈×M₁₆	C₂
socle	C₂²×C₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₇	C₂²×Q₈
cosocle	C₂²×C₄×C₇	C₂⁴
hypercenter	Q₈×M₁₆	D₇
hypocenter	C₇	#22746
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
112. C₇×Q₈	#108	semidirect
209. Q₈	#6396	semidirect
228. C₇	#22746	semidirect

Maximal subgroups

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

subgroup	index
C₇×Q₈×M₁₆	2
Q₈×#1136	2
C₂×Q₈×C₇⋊C₈	2
#11728274	2
#11728270	2
#11727629	2
C₄×#6396	2
#22746	7