Subgroups — #30774370

Order 17152.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863526	C₂	1	1	yes	maximal, characteristic
2	#22561965	C₂	1	1	yes	maximal, characteristic
3	#22558292	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₃₄	C₄	1	1	yes	characteristic
6	C₂×#14523983	C₂²	1	2	yes	—
7	#16863526	—	2	1	no	—
8	#16863526	—	2	1	no	—
9	#16863526	—	2	1	no	—
10	#16863526	—	2	1	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	1	yes	characteristic
12	C₂×Dic₆₇⋊C₈	C₄	1	2	yes	—
13	C₂×#14523953	C₂²	1	2	yes	—
14	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
15	C₄⋊(C₆₇⋊C₁₆)	—	2	1	no	—
16	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	#346×C₆₇	—	2	1	no	—
19	C₄×C₁₆×C₆₇	—	2	1	no	—
20	C₂²×Dic₁₃₄	C₈	1	1	yes	characteristic
21	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₃₄	—	2	1	no	—
23	C₄×Dic₁₃₄	—	2	1	no	—
24	C₄×Dic₁₃₄	—	2	1	no	—
25	C₄×Dic₁₃₄	—	2	1	no	—
26	C₂×C₄⋊Dic₆₇	C₈	1	1	yes	characteristic
27	C₂×Dic₆₇⋊C₄	C₈	1	2	yes	—
28	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
29	#14523983	—	2	4	no	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
33	Dic₆₇⋊C₈	—	2	2	no	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	#14523953	—	2	2	no	—
36	#14523953	—	2	2	no	—
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₆₇	—	2	2	no	—
41	M₃₂×C₆₇	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₆₇	—	2	2	no	—
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₂×Dic₁₃₄	—	2	1	no	—
48	C₂×Dic₁₃₄	—	2	1	no	—
49	C₂×Dic₁₃₄	—	4	2	no	—
50	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
51	C₂²×Dic₆₇	C₂×C₈	1	2	yes	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	—	2	4	no	—
54	C₄⋊Dic₆₇	—	2	1	no	—
55	C₄⋊Dic₆₇	—	2	1	no	—
56	Dic₆₇⋊C₄	—	2	2	no	—
57	Dic₆₇⋊C₄	—	2	2	no	—
58	C₄×Dic₆₇	—	2	2	no	—
59	C₄×Dic₆₇	—	2	2	no	—
60	C₆₇⋊C₁₆	—	4	2	no	—
61	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₆₇	#41	1	2	yes	—
71	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
74	C₄²×C₆₇	M₁₆	1	1	yes	characteristic
75	C₁₆×C₆₇	—	2	4	no	—
76	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	4	1	no	—
79	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#131	1	2	yes	—
87	C₂×Dic₆₇	—	2	2	no	—
88	C₂×Dic₆₇	—	2	2	no	—
89	C₂×Dic₆₇	—	4	2	no	—
90	C₂×Dic₆₇	—	4	2	no	—
91	Dic₁₃₄	—	4	2	no	—
92	Dic₁₃₄	—	4	2	no	—
93	C₈×C₆₇	—	2	1	no	—
94	C₈×C₆₇	—	2	1	no	—
95	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
96	C₈×C₆₇	#124	1	1	yes	characteristic
97	C₈×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
98	C₆₇⋊C₈	—	4	2	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₆₇	C₈×Q₈	1	2	yes	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	#349	1	2	yes	—
107	C₄×C₆₇	#391	1	1	yes	characteristic
108	C₄×C₆₇	#408	1	1	yes	characteristic
109	C₄×C₆₇	#446	1	2	yes	—
110	Dic₆₇	—	4	2	no	—
111	Dic₆₇	—	4	2	no	—
112	#13268	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	2	no	—
114	C₂×C₆₇	#1984	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#349	1	1	yes	characteristic
116	C₂×C₆₇	#2115	1	1	yes	characteristic
117	C₂×#446	—	67	1	no	—
118	#2150	—	67	1	no	—
119	C₄×M₃₂	D₆₇	1	1	yes	characteristic
120	#1377	—	67	4	no	—
121	C₆₇	#13268	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	67	1	no	—
123	C₂×M₃₂	—	67	2	no	—
124	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
125	#446	—	134	1	no	—
126	#446	—	134	1	no	—
127	#446	—	134	1	no	—
128	#446	—	134	1	no	—
129	C₂×C₄⋊C₈	—	67	2	no	—
130	C₂×C₄⋊C₈	—	67	1	no	—
131	C₂×#97	—	67	2	no	—
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₄⋊C₁₆	—	134	1	no	—
134	C₄⋊C₁₆	—	134	1	no	—
135	#346	—	2	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₂²×Q₈	—	67	1	no	—
138	C₂²×C₈	—	67	2	no	—
139	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
141	C₄×Q₈	—	134	1	no	—
142	C₄×Q₈	—	134	1	no	—
143	C₄×Q₈	—	134	1	no	—
144	C₄×Q₈	—	134	1	no	—
145	C₂×C₄⋊C₄	—	67	2	no	—
146	C₂×C₄⋊C₄	—	67	1	no	—
147	C₂×C₄²	—	67	2	no	—
148	C₂×C₄²	C₄×D₆₇	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	134	4	no	—
151	M₃₂	Dic₁₃₄	1	4	yes	complemented
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	134	2	no	—
154	C₄⋊C₈	—	134	2	no	—
155	C₄⋊C₈	—	134	2	no	—
156	C₄⋊C₈	—	134	1	no	—
157	C₄⋊C₈	—	134	1	no	—
158	#97	—	134	2	no	—
159	#97	—	134	2	no	—
160	C₄×C₈	D₂₆₈	1	1	yes	characteristic
161	C₄×C₈	D₂₆₈	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂×Q₈	—	268	2	no	—
165	C₂×Q₈	—	134	1	no	—
166	C₂×Q₈	—	134	1	no	—
167	C₂²×C₄	—	67	2	no	—
168	C₂²×C₄	—	67	2	no	—
169	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	134	2	no	—
174	C₂×C₈	—	134	4	no	—
175	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
178	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
179	C₂×C₈	C₂×Dic₁₃₄	1	2	yes	—
180	C₂×C₈	#11761628	1	2	yes	—
181	C₄⋊C₄	—	134	2	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₄⋊C₄	—	134	1	no	—
184	C₄⋊C₄	—	134	1	no	—
185	C₄²	—	134	2	no	—
186	C₄²	—	134	2	no	—
187	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
188	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
189	C₄²	#11761611	1	1	yes	characteristic
190	C₄²	#11761611	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₁₆	—	268	2	no	—
193	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
194	Q₈	—	268	2	no	—
195	Q₈	—	268	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	268	2	no	—
199	C₂×C₄	—	268	2	no	—
200	C₂×C₄	—	134	2	no	—
201	C₂×C₄	—	134	2	no	—
202	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
203	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
204	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
205	C₂×C₄	(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
206	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
207	C₂×C₄	C₄×Dic₁₃₄	1	2	yes	—
208	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
209	C₂×C₄	#14524049	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	268	2	no	—
213	C₈	#14524022	1	2	yes	—
214	C₈	#14524031	1	1	yes	characteristic
215	C₈	#14524043	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₃₄	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	268	2	no	—
224	C₄	—	268	2	no	—
225	C₄	#16863177	1	2	yes	—
226	C₄	#16863302	1	1	yes	characteristic
227	C₄	#16863362	1	1	yes	characteristic
228	C₄	#16863526	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#22561145	1	1	yes	characteristic
231	C₂	C₂×#16863177	1	1	yes	characteristic
232	C₂	#22561829	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄
commutator, prime radical	C₂³×C₆₇	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₄×M₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₄×M₃₂	D₆₇
hypocenter	C₆₇	#13268
LCS	G ▷ C₂³×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
41. M₃₂×C₆₇	Q₈	semidirect
121. C₆₇	#13268	semidirect
151. M₃₂	Dic₁₃₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863526	2
#22561965	2
#22558292	2
C₄×M₃₂×C₆₇	2
#13268	67