Subgroups — #30874833

Order 17152.

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	Q₈×#11761612	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22560984	C₂	1	3	yes	maximal
4	#22560982	C₂	1	3	yes	maximal
5	#22560346	C₂	1	3	yes	maximal
6	C₄×#14523978	C₂	1	3	yes	maximal
7	Q₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#16863527	—	2	3	no	—
9	Q₈×C₆₇⋊C₈	—	2	1	no	—
10	Q₈×C₆₇⋊C₈	C₄	1	6	yes	—
11	Q₈×C₆₇⋊C₈	C₄	1	2	yes	—
12	C₂×#14523978	C₂²	1	3	yes	—
13	C₂×#14523978	C₂²	1	3	yes	—
14	#16863420	C₂²	1	3	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
16	C₄×#11761612	C₂²	1	3	yes	—
17	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
18	#16863160	C₂²	1	3	yes	—
19	#16863160	C₂²	1	6	yes	—
20	#16863160	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	#446×C₆₇	—	2	3	no	—
23	C₈×Q₈×C₆₇	—	2	1	no	—
24	#423×C₆₇	C₂²	1	3	yes	—
25	C₄×M₁₆×C₆₇	C₂²	1	3	yes	—
26	C₂×#11761612	C₂³	1	3	yes	—
27	C₂×#11761612	C₂³	1	1	yes	characteristic
28	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
29	C₂²×C₆₇⋊C₈	C₂³	1	3	yes	—
30	#14523978	—	2	6	no	—
31	#14523978	—	2	6	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	3	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	3	no	—
34	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	6	yes	—
35	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	6	yes	—
36	#14523954	—	2	3	no	—
37	C₄×C₆₇⋊C₈	—	2	3	no	—
38	C₄×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
39	C₄×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
40	C₂²×Q₈×C₆₇	Q₈	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	C₂³	1	3	yes	—
42	C₂×M₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
44	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	D₄	1	3	yes	—
47	C₂×C₄⋊C₄×C₆₇	Q₈	1	3	yes	—
48	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
49	C₄⋊C₈×C₆₇	—	2	3	no	—
50	C₄⋊C₈×C₆₇	—	2	3	no	—
51	#97×C₆₇	—	2	3	no	—
52	C₄×C₈×C₆₇	—	2	3	no	—
53	#11761612	—	2	1	no	—
54	#11761612	—	2	3	no	—
55	#11761612	—	2	3	no	—
56	#11761612	#41	1	2	yes	—
57	C₂×C₆₇⋊C₈	—	2	1	no	—
58	C₂×C₆₇⋊C₈	—	2	3	no	—
59	C₂×C₆₇⋊C₈	—	2	3	no	—
60	C₂×C₆₇⋊C₈	—	2	1	no	—
61	C₂×C₆₇⋊C₈	—	2	1	no	—
62	C₂×C₆₇⋊C₈	—	2	3	no	—
63	C₂×C₆₇⋊C₈	C₂²×C₄	1	2	yes	—
64	C₂×C₆₇⋊C₈	C₂²×C₄	1	6	yes	—
65	C₂×Q₈×C₆₇	—	2	4	no	—
66	C₂×Q₈×C₆₇	C₄⋊C₄	1	3	yes	—
67	C₂×Q₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
69	C₂²×C₄×C₆₇	C₂×Q₈	1	3	yes	—
70	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
71	M₁₆×C₆₇	—	2	1	no	—
72	M₁₆×C₆₇	—	2	3	no	—
73	M₁₆×C₆₇	—	2	3	no	—
74	M₁₆×C₆₇	#41	1	2	yes	—
75	C₂×C₈×C₆₇	—	2	1	no	—
76	C₂×C₈×C₆₇	—	2	3	no	—
77	C₄⋊C₄×C₆₇	C₄⋊C₄	1	6	yes	—
78	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
79	C₄⋊C₄×C₆₇	C₄⋊C₄	1	3	yes	—
80	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
81	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
82	C₄²×C₆₇	C₂×D₄	1	3	yes	—
83	C₄²×C₆₇	C₂×D₄	1	3	yes	—
84	C₂³×C₆₇	C₂²×Q₈	1	1	yes	characteristic
85	Q₈×C₆₇	—	2	6	no	—
86	Q₈×C₆₇	#108	1	4	yes	complemented
87	C₂×C₄×C₆₇	—	2	6	no	—
88	C₂×C₄×C₆₇	—	4	3	no	—
89	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
92	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
93	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
97	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
98	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
99	C₈×C₆₇	—	2	2	no	—
100	C₈×C₆₇	—	4	3	no	—
101	C₆₇⋊C₈	—	2	6	no	—
102	C₆₇⋊C₈	—	2	2	no	—
103	C₆₇⋊C₈	—	2	2	no	—
104	C₆₇⋊C₈	—	4	3	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₆₇	#557	1	1	yes	characteristic
109	C₂²×C₆₇	#557	1	1	yes	characteristic
110	C₄×C₆₇	—	2	3	no	—
111	C₄×C₆₇	—	2	3	no	—
112	C₄×C₆₇	—	2	3	no	—
113	C₄×C₆₇	C₂×#108	1	6	yes	—
114	C₄×C₆₇	C₄×#41	1	1	yes	characteristic
115	C₄×C₆₇	#519	1	1	yes	characteristic
116	C₄×C₆₇	#548	1	1	yes	characteristic
117	C₄×C₆₇	#548	1	1	yes	characteristic
118	#22746	—	67	1	no	maximal, Sylow
119	C₂×C₆₇	—	2	1	no	—
120	C₂×C₆₇	—	2	1	no	—
121	C₂×C₆₇	#2351	1	1	yes	characteristic
122	C₂×C₆₇	C₂²×#108	1	1	yes	characteristic
123	C₂×C₆₇	#2916	1	1	yes	characteristic
124	Q₈×M₁₆	—	67	1	no	—
125	Q₈×M₁₆	D₆₇	1	1	yes	characteristic
126	C₂×C₈×Q₈	—	67	1	no	—
127	#1950	—	67	3	no	—
128	#1950	—	67	3	no	—
129	#1837	—	67	3	no	—
130	C₄×#108	—	67	3	no	—
131	C₆₇	#22746	1	1	yes	characteristic, complemented, Sylow
132	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
133	#446	—	2	3	no	—
134	#446	—	134	3	no	—
135	C₈×Q₈	—	2	1	no	—
136	C₈×Q₈	—	67	6	no	—
137	C₈×Q₈	—	67	2	no	—
138	C₈×Q₈	—	134	1	no	—
139	C₂×#108	—	67	3	no	—
140	C₂×#108	—	67	3	no	—
141	#423	—	67	3	no	—
142	#423	C₂×D₆₇	1	3	yes	—
143	C₂×C₄⋊C₈	—	67	3	no	—
144	C₄×M₁₆	—	67	3	no	—
145	C₄×M₁₆	C₂×D₆₇	1	3	yes	—
146	C₂×C₄×C₈	—	67	3	no	—
147	#341	—	67	3	no	—
148	#341	—	67	6	no	—
149	#341	—	67	1	no	—
150	C₂²×Q₈	Dic₁₃₄	1	1	yes	characteristic
151	C₂×M₁₆	—	67	1	no	—
152	C₂×M₁₆	—	67	3	no	—
153	C₂×M₁₆	C₂²×D₆₇	1	3	yes	—
154	C₂×M₁₆	C₂²×D₆₇	1	1	yes	characteristic
155	C₂²×C₈	—	67	1	no	—
156	C₂²×C₈	—	67	3	no	—
157	C₄×Q₈	C₄×D₆₇	1	3	yes	—
158	C₄×Q₈	C₄×D₆₇	1	1	yes	characteristic
159	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
160	C₄×Q₈	C₂²⋊D₆₇	1	3	yes	—
161	C₂×C₄⋊C₄	Dic₁₃₄	1	3	yes	—
162	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
163	#108	—	134	6	no	—
164	#108	—	134	6	no	—
165	C₄⋊C₈	—	2	3	no	—
166	C₄⋊C₈	—	2	3	no	—
167	C₄⋊C₈	—	67	6	no	—
168	C₄⋊C₈	—	67	6	no	—
169	C₄⋊C₈	—	134	3	no	—
170	C₄⋊C₈	—	134	3	no	—
171	#97	—	2	3	no	—
172	#97	—	134	3	no	—
173	C₄×C₈	—	2	3	no	—
174	C₄×C₈	—	67	6	no	—
175	C₄×C₈	—	67	6	no	—
176	C₄×C₈	—	134	3	no	—
177	C₂×Q₈	—	2	4	no	—
178	C₂×Q₈	Dic₆₇⋊C₄	1	3	yes	—
179	C₂×Q₈	Dic₆₇⋊C₄	1	1	yes	characteristic
180	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
181	C₂²×C₄	C₂×Dic₁₃₄	1	3	yes	—
182	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
183	M₁₆	—	2	1	no	—
184	M₁₆	—	2	3	no	—
185	M₁₆	—	2	3	no	—
186	M₁₆	—	67	2	no	—
187	M₁₆	—	134	3	no	—
188	M₁₆	—	134	3	no	—
189	M₁₆	—	134	1	no	—
190	M₁₆	#11761629	1	2	yes	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	3	no	—
193	C₂×C₈	—	67	2	no	—
194	C₂×C₈	—	67	6	no	—
195	C₂×C₈	—	134	3	no	—
196	C₂×C₈	—	134	1	no	—
197	C₂×C₈	—	134	3	no	—
198	C₂×C₈	—	134	3	no	—
199	C₂×C₈	—	134	1	no	—
200	C₂×C₈	—	134	1	no	—
201	C₄⋊C₄	Dic₆₇⋊C₄	1	6	yes	—
202	C₄⋊C₄	Dic₆₇⋊C₄	1	3	yes	—
203	C₄⋊C₄	Dic₆₇⋊C₄	1	3	yes	—
204	C₄²	C₂×C₄×D₆₇	1	3	yes	—
205	C₄²	C₂×C₄×D₆₇	1	3	yes	—
206	C₄²	C₂×C₂²⋊D₆₇	1	3	yes	—
207	C₄²	C₂×C₂²⋊D₆₇	1	3	yes	—
208	C₂³	C₂²×Dic₁₃₄	1	1	yes	characteristic
209	Q₈	—	2	6	no	—
210	Q₈	#14523978	1	4	yes	complemented
211	C₂×C₄	—	2	6	no	—
212	C₂×C₄	—	4	3	no	—
213	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
214	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
215	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
216	C₂×C₄	C₂×Dic₆₇⋊C₄	1	3	yes	—
217	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
218	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
219	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
220	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
221	C₂×C₄	#14524090	1	1	yes	characteristic
222	C₂×C₄	#14524091	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	3	no	—
225	C₈	—	268	3	no	—
226	C₈	—	134	2	no	—
227	C₈	—	134	6	no	—
228	C₈	—	134	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	C₂²×Dic₆₇⋊C₄	1	1	yes	characteristic
232	C₂²	#16864122	1	1	yes	characteristic
233	C₂²	#16864125	1	1	yes	characteristic
234	C₄	—	2	3	no	—
235	C₄	—	2	3	no	—
236	C₄	—	2	3	no	—
237	C₄	C₂×#14523978	1	6	yes	—
238	C₄	C₄×#11761629	1	1	yes	characteristic
239	C₄	#16863872	1	1	yes	characteristic
240	C₄	#16864053	1	1	yes	characteristic
241	C₄	#16864057	1	1	yes	characteristic
242	C₂	—	2	1	no	—
243	C₂	—	2	1	no	—
244	C₂	#22563127	1	1	yes	characteristic
245	C₂	C₂²×#14523978	1	1	yes	characteristic
246	C₂	#22568818	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	Q₈×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×C₆₇	C₂²×Q₈
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	Q₈×M₁₆	D₆₇
hypocenter	C₆₇	#22746
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
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
86. Q₈×C₆₇	#108	semidirect
131. C₆₇	#22746	semidirect
210. Q₈	#14523978	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761612	2
C₂×Q₈×C₆₇⋊C₈	2
#22560984	2
#22560982	2
#22560346	2
C₄×#14523978	2
Q₈×M₁₆×C₆₇	2
#22746	67