Subgroups — C₂×#22569298 #31720780

Order 17152.

2022 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569298	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	6	yes	maximal, complemented
3	C₂×Q₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
4	C₂×#16863453	C₂	1	3	yes	maximal
5	C₂×#16863449	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#16863527	C₂²	1	48	yes	complemented
8	Q₈×C₆₇⋊C₈	C₂²	1	16	yes	complemented
9	#16863453	C₂²	1	24	yes	—
10	#16863449	C₂²	1	24	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	6	yes	—
13	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	3	yes	—
15	C₂×#14523954	C₂²	1	6	yes	—
16	C₂×C₄×C₆₇⋊C₈	C₂²	1	6	yes	—
17	C₂³×Q₈×C₆₇	C₄	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	C₄	1	8	yes	—
19	C₂×C₄×Q₈×C₆₇	C₄	1	3	yes	—
20	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₂²	1	8	yes	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	3	yes	—
24	C₂²×C₄⋊C₄×C₆₇	C₄	1	3	yes	—
25	C₂²×C₄²×C₆₇	C₂²	1	3	yes	—
26	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
27	C₂²×C₆₇⋊C₈	C₂³	1	6	yes	—
28	C₄⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
29	C₄⋊(C₆₇⋊C₈)	C₂³	1	24	yes	—
30	C₂²⋊(C₆₇⋊C₈)	C₂³	1	4	yes	—
31	C₂²⋊(C₆₇⋊C₈)	C₂³	1	12	yes	—
32	#14523954	C₂³	1	24	yes	—
33	C₄×C₆₇⋊C₈	C₂³	1	24	yes	—
34	C₂²×Q₈×C₆₇	—	2	8	no	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	8	yes	—
36	C₂²×Q₈×C₆₇	C₂×C₄	1	3	yes	—
37	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
39	C₂³×C₄×C₆₇	C₂×C₄	1	3	yes	—
40	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
41	C₄×Q₈×C₆₇	C₂×C₄	1	24	yes	—
42	C₄×Q₈×C₆₇	C₂×C₄	1	12	yes	—
43	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	—
44	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
45	C₄×Q₈×C₆₇	C₂³	1	4	yes	—
46	C₄×Q₈×C₆₇	C₂³	1	12	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	24	yes	—
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	6	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	3	yes	—
51	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	12	yes	—
53	C₂×C₄²×C₆₇	C₂×C₄	1	3	yes	—
54	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
55	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
56	C₂×C₄²×C₆₇	C₂³	1	12	yes	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	12	no	—
59	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
60	C₂×C₆₇⋊C₈	C₂⁴	1	12	yes	—
61	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂×Q₈×C₆₇	—	2	32	no	—
63	C₂×Q₈×C₆₇	—	2	12	no	—
64	C₂×Q₈×C₆₇	M₁₆	1	8	yes	—
65	C₂×Q₈×C₆₇	C₂²×C₄	1	12	yes	—
66	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
67	C₂²×C₄×C₆₇	—	2	12	no	—
68	C₂²×C₄×C₆₇	—	2	12	no	—
69	C₂²×C₄×C₆₇	—	2	4	no	—
70	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
72	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
73	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
74	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	12	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
78	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
80	C₄⋊C₄×C₆₇	C₂²×C₄	1	24	yes	—
81	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
82	C₄⋊C₄×C₆₇	C₂²×C₄	1	12	yes	—
83	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
84	C₄²×C₆₇	C₂²×C₄	1	12	yes	—
85	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
86	C₄²×C₆₇	C₂²×C₄	1	6	yes	—
87	C₄²×C₆₇	C₂⁴	1	6	yes	—
88	C₄²×C₆₇	C₂⁴	1	6	yes	—
89	C₂³×C₆₇	—	2	2	no	—
90	C₂³×C₆₇	—	2	2	no	—
91	C₂³×C₆₇	#131	1	4	yes	—
92	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
93	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₆₇	—	2	24	no	—
95	Q₈×C₆₇	C₂×M₁₆	1	16	yes	—
96	C₂×C₄×C₆₇	—	2	2	no	—
97	C₂×C₄×C₆₇	—	2	6	no	—
98	C₂×C₄×C₆₇	—	2	24	no	—
99	C₂×C₄×C₆₇	—	2	6	no	—
100	C₂×C₄×C₆₇	—	2	6	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	24	no	—
103	C₂×C₄×C₆₇	—	2	4	no	—
104	C₂×C₄×C₆₇	C₂×M₁₆	1	12	yes	—
105	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
106	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
108	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
110	C₂×C₄×C₆₇	C₂³×C₄	1	6	yes	—
111	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
112	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
113	C₆₇⋊C₈	—	2	4	no	—
114	C₆₇⋊C₈	—	2	12	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	8	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	#519	1	2	yes	—
119	C₂²×C₆₇	C₂²×M₁₆	1	2	yes	—
120	C₂²×C₆₇	C₂×#131	1	4	yes	—
121	C₂²×C₆₇	C₂×#131	1	2	yes	—
122	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
123	C₄×C₆₇	—	2	2	no	—
124	C₄×C₆₇	—	2	6	no	—
125	C₄×C₆₇	—	2	6	no	—
126	C₄×C₆₇	—	2	6	no	—
127	C₄×C₆₇	#519	1	2	yes	—
128	C₄×C₆₇	C₂²×M₁₆	1	12	yes	—
129	C₄×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
130	C₂×#2984	—	67	1	no	maximal, Sylow
131	C₂×C₆₇	—	2	4	no	—
132	C₂×C₆₇	#2984	1	4	yes	complemented
133	C₂×C₆₇	C₂×#519	1	1	yes	characteristic
134	C₂×C₆₇	C₂³×M₁₆	1	1	yes	characteristic
135	C₂×C₆₇	C₂²×#131	1	1	yes	characteristic
136	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
137	#2984	—	67	16	no	—
138	C₂×#446	—	67	6	no	—
139	C₂×C₈×Q₈	—	67	2	no	—
140	C₂×#432	—	67	3	no	—
141	C₂×#431	—	67	3	no	—
142	C₆₇	C₂×#2984	1	1	yes	characteristic, complemented, Sylow
143	C₂³×Q₈	Dic₆₇	1	1	yes	characteristic
144	C₂×C₄×Q₈	Dic₆₇	1	8	yes	—
145	C₂×C₄×Q₈	Dic₆₇	1	3	yes	—
146	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
147	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
148	C₂×C₄×Q₈	C₂×D₆₇	1	3	yes	—
149	C₂×C₄×Q₈	C₂×D₆₇	1	8	yes	—
150	C₂²×C₄⋊C₄	Dic₆₇	1	3	yes	—
151	C₂²×C₄²	C₂×D₆₇	1	3	yes	—
152	#446	—	67	48	no	—
153	C₈×Q₈	—	67	16	no	—
154	#432	—	67	24	no	—
155	#431	—	67	24	no	—
156	C₂×C₄⋊C₈	—	67	6	no	—
157	C₂×C₄⋊C₈	—	67	6	no	—
158	C₂×C₂²⋊C₈	—	67	1	no	—
159	C₂×C₂²⋊C₈	—	67	3	no	—
160	C₂×#97	—	67	6	no	—
161	C₂×C₄×C₈	—	67	6	no	—
162	C₂²×Q₈	—	2	8	no	—
163	C₂²×Q₈	C₂×Dic₆₇	1	8	yes	—
164	C₂²×Q₈	C₂×Dic₆₇	1	3	yes	—
165	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
166	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
167	C₂³×C₄	C₂×Dic₆₇	1	3	yes	—
168	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
169	C₂²×C₈	—	67	2	no	—
170	C₂²×C₈	—	67	6	no	—
171	C₄×Q₈	C₂×Dic₆₇	1	12	yes	—
172	C₄×Q₈	C₂×Dic₆₇	1	24	yes	—
173	C₄×Q₈	C₂×Dic₆₇	1	4	yes	—
174	C₄×Q₈	C₂×Dic₆₇	1	8	yes	—
175	C₄×Q₈	C₂²×D₆₇	1	4	yes	—
176	C₄×Q₈	C₂²×D₆₇	1	12	yes	—
177	C₂×C₄⋊C₄	C₂×Dic₆₇	1	6	yes	—
178	C₂×C₄⋊C₄	C₂×Dic₆₇	1	24	yes	—
179	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
180	C₂×C₄⋊C₄	C₂×Dic₆₇	1	3	yes	—
181	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
182	C₂×C₄²	C₂×Dic₆₇	1	12	yes	—
183	C₂×C₄²	C₂×Dic₆₇	1	3	yes	—
184	C₂×C₄²	C₂²×D₆₇	1	12	yes	—
185	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
186	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
187	C₄⋊C₈	—	67	24	no	—
188	C₄⋊C₈	—	67	24	no	—
189	C₂²⋊C₈	—	67	4	no	—
190	C₂²⋊C₈	—	67	12	no	—
191	#97	—	67	24	no	—
192	C₄×C₈	—	67	24	no	—
193	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
194	C₂×Q₈	—	2	32	no	—
195	C₂×Q₈	—	2	12	no	—
196	C₂×Q₈	#11761612	1	8	yes	—
197	C₂×Q₈	C₂²×Dic₆₇	1	12	yes	—
198	C₂×Q₈	C₂²×Dic₆₇	1	4	yes	—
199	C₂²×C₄	—	2	12	no	—
200	C₂²×C₄	—	2	12	no	—
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
204	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
205	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
206	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
207	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
208	C₂²×C₄	C₂²×Dic₆₇	1	12	yes	—
209	C₂²×C₄	C₂²×Dic₆₇	1	3	yes	—
210	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
211	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
212	C₂×C₈	—	67	4	no	—
213	C₂×C₈	—	67	12	no	—
214	C₂×C₈	—	134	4	no	—
215	C₂×C₈	—	134	12	no	—
216	C₄⋊C₄	C₂²×Dic₆₇	1	24	yes	—
217	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
218	C₄⋊C₄	C₂²×Dic₆₇	1	12	yes	—
219	C₄²	C₂²×Dic₆₇	1	12	yes	—
220	C₄²	C₂²×Dic₆₇	1	12	yes	—
221	C₄²	C₂²×Dic₆₇	1	6	yes	—
222	C₄²	C₂²×Dic₆₇	1	6	yes	—
223	C₄²	C₂³×D₆₇	1	6	yes	—
224	C₄²	C₂³×D₆₇	1	6	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	#14524050	1	4	yes	—
228	C₂³	C₂³×Dic₆₇	1	2	yes	—
229	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
230	Q₈	—	2	24	no	—
231	Q₈	C₂×#11761612	1	16	yes	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	6	no	—
234	C₂×C₄	—	2	24	no	—
235	C₂×C₄	—	2	6	no	—
236	C₂×C₄	—	2	6	no	—
237	C₂×C₄	—	2	4	no	—
238	C₂×C₄	—	2	24	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	C₂×#11761612	1	12	yes	—
241	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
242	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
243	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
244	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
245	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
246	C₂×C₄	C₂³×Dic₆₇	1	6	yes	—
247	C₂×C₄	#14524091	1	4	yes	—
248	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
249	C₈	—	134	4	no	—
250	C₈	—	134	12	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	2	no	—
254	C₂²	#16863874	1	2	yes	—
255	C₂²	C₂²×#11761612	1	2	yes	—
256	C₂²	C₂×#14524050	1	4	yes	—
257	C₂²	C₂×#14524050	1	2	yes	—
258	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
259	C₄	—	2	6	no	—
260	C₄	—	2	6	no	—
261	C₄	—	2	6	no	—
262	C₄	—	2	2	no	—
263	C₄	#16863874	1	2	yes	—
264	C₄	C₂²×#11761612	1	12	yes	—
265	C₄	C₂×#14524091	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	#22569298	1	4	yes	direct product factor
268	C₂	C₂×#16863874	1	1	yes	characteristic
269	C₂	C₂³×#11761612	1	1	yes	characteristic
270	C₂	C₂²×#14524050	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂×#2984
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. #22569298	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
3. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
7. #16863527	C₂²	semidirect
8. Q₈×C₆₇⋊C₈	C₂²	semidirect
132. C₂×C₆₇	#2984	semidirect
142. C₆₇	C₂×#2984	semidirect
267. C₂	#22569298	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569298	2
C₂×#16863527	2
C₂×Q₈×C₆₇⋊C₈	2
C₂×#16863453	2
C₂×#16863449	2
C₂²×C₄×Q₈×C₆₇	2
C₂×#2984	67

Subgroups — C2×#22569298 #31720780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569298 #31720780