Subgroups — #30746493

Order 17152.

498 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	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	#22558050	C₂	1	2	yes	maximal
5	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863151	—	2	1	no	—
10	#16863151	—	2	2	no	—
11	Q₈⋊(C₆₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#391×C₆₇	C₄	1	2	yes	—
14	#389×C₆₇	C₄	1	2	yes	—
15	#387×C₆₇	—	2	2	no	—
16	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
18	#376×C₆₇	—	2	1	no	—
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×C₆₇⋊C₈	—	2	2	no	—
21	C₂²×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₄×C₆₇⋊C₈	—	2	2	no	—
25	C₄×C₆₇⋊C₈	—	4	2	no	—
26	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×Q₈×C₆₇	—	2	2	no	—
29	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
31	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	—	2	1	no	—
35	C₂×C₄²×C₆₇	—	2	1	no	—
36	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	#95×C₆₇	—	2	2	no	—
42	#95×C₆₇	—	2	2	no	—
43	#95×C₆₇	—	2	2	no	—
44	#95×C₆₇	—	2	1	no	—
45	C₂×C₆₇⋊C₈	—	2	4	no	—
46	C₂×C₆₇⋊C₈	—	2	4	no	—
47	C₂×C₆₇⋊C₈	—	4	4	no	—
48	C₂×C₆₇⋊C₈	—	4	2	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	2	1	no	—
51	C₂×Q₈×C₆₇	—	2	4	no	—
52	C₂×Q₈×C₆₇	C₄²	1	2	yes	—
53	C₂²×C₄×C₆₇	—	2	1	no	—
54	C₂²×C₄×C₆₇	—	2	1	no	—
55	C₂²×C₄×C₆₇	—	2	2	no	—
56	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	1	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	—	2	4	no	—
64	C₄⋊C₄×C₆₇	—	4	4	no	—
65	C₄⋊C₄×C₆₇	C₄²	1	2	yes	—
66	C₄²×C₆₇	—	2	4	no	—
67	C₄²×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	2	2	no	—
69	C₄²×C₆₇	—	4	2	no	—
70	C₄²×C₆₇	—	4	2	no	—
71	C₄²×C₆₇	SD₁₆	1	2	yes	—
72	C₄²×C₆₇	Q₁₆	1	2	yes	—
73	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	2	4	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	#103	1	4	yes	—
92	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
93	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
96	C₂×C₄×C₆₇	#131	1	4	yes	—
97	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#135	1	2	yes	—
100	C₆₇⋊C₈	—	4	4	no	—
101	C₆₇⋊C₈	—	4	4	no	—
102	C₂²×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
103	C₂²×C₆₇	#408	1	2	yes	—
104	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
105	C₂²×C₆₇	#416	1	1	yes	characteristic
106	C₂²×C₆₇	#421	1	2	yes	—
107	C₄×C₆₇	—	2	4	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	4	2	no	—
110	C₄×C₆₇	—	4	2	no	—
111	C₄×C₆₇	—	4	2	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#405	1	2	yes	—
114	C₄×C₆₇	#433	1	2	yes	—
115	C₄×C₆₇	C₄×SD₁₆	1	2	yes	—
116	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
117	#8182	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	#1481	1	2	yes	—
119	C₂×C₆₇	#1482	1	2	yes	—
120	C₂×C₆₇	#1750	1	1	yes	characteristic
121	C₂×C₆₇	#1759	1	1	yes	characteristic
122	C₂×C₆₇	C₄×#103	1	1	yes	characteristic
123	#2295	D₆₇	1	1	yes	characteristic
124	C₄×C₄⋊C₈	—	67	1	no	—
125	#1744	—	67	1	no	—
126	C₂×Q₈⋊C₈	—	67	2	no	—
127	#1279	—	67	2	no	—
128	C₆₇	#8182	1	1	yes	characteristic, complemented, Sylow
129	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
130	C₂×C₄⋊C₈	—	67	2	no	—
131	C₂×C₄×C₈	—	67	2	no	—
132	C₂×C₄×C₈	—	134	1	no	—
133	#391	Dic₆₇	1	2	yes	—
134	#389	Dic₆₇	1	2	yes	—
135	#387	—	2	2	no	—
136	C₄×C₄⋊C₄	—	2	1	no	—
137	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
138	#376	—	2	1	no	—
139	C₄³	C₂×D₆₇	1	1	yes	characteristic
140	#336	—	134	2	no	—
141	#336	—	134	1	no	—
142	Q₈⋊C₈	—	67	8	no	—
143	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₈	—	134	2	no	—
145	C₂²×C₈	—	134	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	C₄×D₆₇	1	4	yes	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
152	C₂×C₄²	—	2	1	no	—
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
155	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
156	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
157	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊C₈	—	67	4	no	—
160	C₄⋊C₈	—	134	2	no	—
161	C₄×C₈	—	268	2	no	—
162	C₄×C₈	—	67	4	no	—
163	C₄×C₈	—	134	2	no	—
164	#95	—	2	2	no	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	1	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	4	no	—
170	C₂×Q₈	C₄×Dic₆₇	1	2	yes	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
176	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂×C₈	—	268	4	no	—
180	C₂×C₈	—	268	2	no	—
181	C₂×C₈	—	268	2	no	—
182	C₂×C₈	—	134	4	no	—
183	C₂×C₈	—	134	4	no	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	4	no	—
188	C₄⋊C₄	C₄×Dic₆₇	1	2	yes	—
189	C₄²	—	2	4	no	—
190	C₄²	—	2	4	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	2	no	—
193	C₄²	—	4	2	no	—
194	C₄²	#11761615	1	2	yes	—
195	C₄²	#11761618	1	2	yes	—
196	C₄²	C₂×C₄×D₆₇	1	2	yes	—
197	C₄²	#11761628	1	2	yes	—
198	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
199	Q₈	—	2	4	no	—
200	Q₈	—	4	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	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	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	#14523967	1	4	yes	—
215	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
216	C₂×C₄	#14524002	1	1	yes	characteristic
217	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
218	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
219	C₂×C₄	#14524050	1	4	yes	—
220	C₂×C₄	C₂×#11761615	1	1	yes	characteristic
221	C₂×C₄	C₂×#11761618	1	1	yes	characteristic
222	C₂×C₄	#14524063	1	2	yes	—
223	C₈	—	268	4	no	—
224	C₈	—	268	4	no	—
225	C₂²	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
226	C₂²	#16863363	1	2	yes	—
227	C₂²	C₂×#14523967	1	1	yes	characteristic
228	C₂²	#16863392	1	1	yes	characteristic
229	C₂²	#16863413	1	2	yes	—
230	C₄	—	2	2	no	—
231	C₄	—	2	4	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	#16863352	1	2	yes	—
237	C₄	#16863457	1	2	yes	—
238	C₄	C₄×#11761615	1	2	yes	—
239	C₄	C₄×#11761618	1	2	yes	—
240	C₂	#22558612	1	2	yes	—
241	C₂	#22558617	1	2	yes	—
242	C₂	#22559975	1	1	yes	characteristic
243	C₂	#22560019	1	1	yes	characteristic
244	C₂	C₄×#14523967	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2295	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#8182
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2295

Complements

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

N	H ≅ G/N	type
11. Q₈⋊(C₆₇⋊C₈)	C₄	semidirect
128. C₆₇	#8182	semidirect

Maximal subgroups

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

subgroup	index
C₄×C₄⋊(C₆₇⋊C₈)	2
#22559955	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
#22558050	2
C₆₇×#2295	2
#8182	67