Subgroups — C₂×#22569150 #31720204

Order 17152.

1894 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22569150	C₂	1	16	yes	maximal, direct product factor
2	C₂×#16863527	C₂	1	8	yes	maximal, complemented
3	C₂×#16863423	C₂	1	2	yes	maximal
4	C₂×#16863420	C₂	1	1	yes	maximal, characteristic
5	C₂×#16863352	C₂	1	2	yes	maximal
6	C₂²×#14523954	C₂	1	1	yes	maximal, characteristic
7	C₂×#533×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11761612	C₂²	1	2	yes	—
9	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
11	#16863527	C₂²	1	64	yes	complemented
12	#16863423	C₂²	1	16	yes	—
13	#16863420	C₂²	1	8	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	8	yes	—
16	#16863352	C₂²	1	16	yes	—
17	C₂×#14523954	C₂²	1	4	yes	—
18	C₂×#14523954	C₂²	1	8	yes	—
19	C₂×#14523954	C₂²	1	4	yes	—
20	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
21	#533×C₆₇	C₄	1	8	yes	—
22	#533×C₆₇	C₂²	1	8	yes	—
23	C₂×C₄⋊Q₈×C₆₇	C₄	1	2	yes	—
24	C₂×#125×C₆₇	C₄	1	2	yes	—
25	C₂×C₂²⋊Q₈×C₆₇	C₄	1	4	yes	—
26	C₂×C₄×Q₈×C₆₇	C₂²	1	4	yes	—
27	C₂×#117×C₆₇	C₂²	1	2	yes	—
28	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×#11761612	—	2	8	no	—
30	C₂×#11761612	C₂³	1	8	yes	—
31	C₂²×C₆₇⋊C₈	—	2	4	no	—
32	C₂²×C₆₇⋊C₈	Q₈	1	8	yes	complemented
33	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
36	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
37	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
38	C₄⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
39	C₄⋊(C₆₇⋊C₈)	C₂³	1	32	yes	—
40	#14523954	C₂³	1	16	yes	—
41	#14523954	C₂³	1	16	yes	—
42	C₄×C₆₇⋊C₈	C₂³	1	16	yes	—
43	C₂²×Q₈×C₆₇	C₂×C₄	1	4	yes	—
44	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₄⋊Q₈×C₆₇	C₂×C₄	1	16	yes	—
47	#125×C₆₇	C₂×C₄	1	16	yes	—
48	C₂²⋊Q₈×C₆₇	C₂×C₄	1	32	yes	—
49	C₄×Q₈×C₆₇	C₂×C₄	1	16	yes	—
50	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
51	#117×C₆₇	C₂×C₄	1	8	yes	—
52	#117×C₆₇	C₂³	1	8	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	—
55	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
56	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	—
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	4	yes	—
63	#11761612	—	2	16	no	—
64	C₂×C₆₇⋊C₈	—	2	4	no	—
65	C₂×C₆₇⋊C₈	—	2	8	no	—
66	C₂×C₆₇⋊C₈	—	2	8	no	—
67	C₂×C₆₇⋊C₈	C₂×Q₈	1	8	yes	—
68	C₂×C₆₇⋊C₈	C₂×Q₈	1	16	yes	complemented
69	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
70	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
71	C₂×C₆₇⋊C₈	C₂⁴	1	4	yes	—
72	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₂×Q₈×C₆₇	—	4	8	no	—
74	C₂×Q₈×C₆₇	C₂²×C₄	1	16	yes	—
75	C₂²×C₄×C₆₇	—	4	4	no	—
76	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
79	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
83	C₂²×C₄×C₆₇	#41	1	4	yes	—
84	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
87	C₄⋊C₄×C₆₇	C₂²×C₄	1	32	yes	—
88	C₄⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
89	C₂²⋊C₄×C₆₇	C₂²×C₄	1	16	yes	—
90	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
91	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
92	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
93	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
94	C₄²×C₆₇	C₂⁴	1	4	yes	—
95	C₄²×C₆₇	C₂⁴	1	8	yes	—
96	C₂³×C₆₇	—	2	4	no	—
97	C₂³×C₆₇	C₄×Q₈	1	4	yes	—
98	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
99	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
100	Q₈×C₆₇	—	4	16	no	—
101	C₂×C₄×C₆₇	—	2	4	no	—
102	C₂×C₄×C₆₇	—	2	4	no	—
103	C₂×C₄×C₆₇	—	2	16	no	—
104	C₂×C₄×C₆₇	—	4	8	no	—
105	C₂×C₄×C₆₇	C₄×Q₈	1	8	yes	—
106	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
107	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
108	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
109	C₂×C₄×C₆₇	C₂³×C₄	1	16	yes	—
110	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
113	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
114	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
115	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
116	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
117	C₆₇⋊C₈	—	2	4	no	—
118	C₆₇⋊C₈	—	2	8	no	—
119	C₆₇⋊C₈	C₂²×Q₈	1	8	yes	—
120	C₂²×C₆₇	—	2	8	no	—
121	C₂²×C₆₇	—	2	4	no	—
122	C₂²×C₆₇	C₂×C₄×Q₈	1	4	yes	—
123	C₂²×C₆₇	C₂×C₄×Q₈	1	2	yes	—
124	C₂²×C₆₇	#568	1	4	yes	—
125	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
126	C₄×C₆₇	—	2	4	no	—
127	C₄×C₆₇	—	2	4	no	—
128	C₄×C₆₇	—	2	16	no	—
129	C₄×C₆₇	C₂×C₄×Q₈	1	2	yes	—
130	C₄×C₆₇	C₂×C₄×Q₈	1	4	yes	—
131	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
132	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
133	C₂×#2961	—	67	1	no	maximal, Sylow
134	C₂×C₆₇	—	2	4	no	—
135	C₂×C₆₇	#2961	1	4	yes	complemented
136	C₂×C₆₇	C₂²×C₄×Q₈	1	1	yes	characteristic
137	C₂×C₆₇	C₂×#568	1	2	yes	—
138	C₂×#533	D₆₇	1	1	yes	characteristic
139	#2961	—	67	16	no	—
140	C₂×#446	—	67	8	no	—
141	C₂×#424	—	67	2	no	—
142	C₂×#423	—	67	1	no	—
143	C₂×#405	—	67	2	no	—
144	C₂²×#97	—	67	1	no	—
145	C₆₇	C₂×#2961	1	1	yes	characteristic, complemented, Sylow
146	C₂²×M₁₆	—	67	2	no	—
147	C₂³×C₈	—	67	1	no	—
148	C₂³×C₈	—	67	1	no	—
149	#533	Dic₆₇	1	8	yes	—
150	#533	C₂×D₆₇	1	8	yes	—
151	C₂×C₄⋊Q₈	Dic₆₇	1	2	yes	—
152	C₂×#125	Dic₆₇	1	2	yes	—
153	C₂×C₂²⋊Q₈	Dic₆₇	1	4	yes	—
154	C₂×C₄×Q₈	C₂×D₆₇	1	4	yes	—
155	C₂×#117	C₂×D₆₇	1	2	yes	—
156	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
157	#446	—	67	64	no	—
158	#424	—	67	16	no	—
159	#423	—	67	8	no	—
160	C₂×C₄⋊C₈	—	67	4	no	—
161	C₂×C₄⋊C₈	—	67	8	no	—
162	#405	—	67	16	no	—
163	C₂×#97	—	67	4	no	—
164	C₂×#97	—	67	8	no	—
165	C₂×#97	—	67	4	no	—
166	C₂×C₄×C₈	—	67	4	no	—
167	C₂²×Q₈	C₂×Dic₆₇	1	4	yes	—
168	C₂³×C₄	C₂×Dic₆₇	1	2	yes	—
169	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
170	C₂×M₁₆	—	67	8	no	—
171	C₂×M₁₆	—	134	8	no	—
172	C₂²×C₈	—	67	8	no	—
173	C₂²×C₈	—	67	4	no	—
174	C₂²×C₈	—	67	2	no	—
175	C₂²×C₈	—	67	4	no	—
176	C₂²×C₈	—	67	2	no	—
177	C₂²×C₈	—	67	4	no	—
178	C₂²×C₈	—	134	4	no	—
179	C₄⋊Q₈	C₂×Dic₆₇	1	16	yes	—
180	#125	C₂×Dic₆₇	1	16	yes	—
181	C₂²⋊Q₈	C₂×Dic₆₇	1	32	yes	—
182	C₄×Q₈	C₂×Dic₆₇	1	16	yes	—
183	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
184	#117	C₂×Dic₆₇	1	8	yes	—
185	#117	C₂²×D₆₇	1	8	yes	—
186	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
187	C₂×C₄⋊C₄	C₂×Dic₆₇	1	8	yes	—
188	C₂×C₄⋊C₄	C₂×Dic₆₇	1	4	yes	—
189	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	4	yes	—
190	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
191	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
192	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
193	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
194	C₂×C₄²	C₂²×D₆₇	1	4	yes	—
195	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
196	C₄⋊C₈	—	67	16	no	—
197	C₄⋊C₈	—	67	32	no	—
198	#97	—	67	16	no	—
199	#97	—	67	16	no	—
200	C₄×C₈	—	67	16	no	—
201	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
202	C₂×Q₈	—	4	8	no	—
203	C₂×Q₈	C₂²×Dic₆₇	1	16	yes	—
204	C₂²×C₄	—	4	4	no	—
205	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
206	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
208	C₂²×C₄	C₂²×Dic₆₇	1	8	yes	—
209	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
211	C₂²×C₄	Q₈×D₆₇	1	4	yes	—
212	C₂²×C₄	#11761629	1	4	yes	—
213	C₂²×C₄	C₂³×D₆₇	1	2	yes	—
214	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
215	M₁₆	—	134	16	no	—
216	C₂×C₈	—	67	8	no	—
217	C₂×C₈	—	67	16	no	—
218	C₂×C₈	—	67	4	no	—
219	C₂×C₈	—	67	8	no	—
220	C₂×C₈	—	67	4	no	—
221	C₂×C₈	—	134	4	no	—
222	C₂×C₈	—	134	8	no	—
223	C₂×C₈	—	134	8	no	—
224	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
225	C₄⋊C₄	C₂²×Dic₆₇	1	32	yes	—
226	C₄⋊C₄	C₂²×Dic₆₇	1	16	yes	—
227	C₂²⋊C₄	C₂²×Dic₆₇	1	16	yes	—
228	C₄²	C₂²×Dic₆₇	1	4	yes	—
229	C₄²	C₂²×Dic₆₇	1	4	yes	—
230	C₄²	C₂²×Dic₆₇	1	4	yes	—
231	C₄²	C₂²×Dic₆₇	1	8	yes	—
232	C₄²	C₂³×D₆₇	1	8	yes	—
233	C₄²	C₂³×D₆₇	1	4	yes	—
234	C₂³	—	2	4	no	—
235	C₂³	Q₈×Dic₆₇	1	4	yes	—
236	C₂³	C₂³×Dic₆₇	1	2	yes	—
237	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
238	Q₈	—	4	16	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	16	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	Q₈×Dic₆₇	1	8	yes	—
244	C₂×C₄	Q₈×Dic₆₇	1	4	yes	—
245	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
246	C₂×C₄	C₂³×Dic₆₇	1	4	yes	—
247	C₂×C₄	C₂³×Dic₆₇	1	16	yes	—
248	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
249	C₂×C₄	C₂³×Dic₆₇	1	2	yes	—
250	C₂×C₄	C₂×Q₈×D₆₇	1	4	yes	—
251	C₂×C₄	C₂×Q₈×D₆₇	1	2	yes	—
252	C₂×C₄	C₂×#11761629	1	2	yes	—
253	C₂×C₄	C₂×#11761629	1	4	yes	—
254	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
255	C₈	—	67	8	no	—
256	C₈	—	134	4	no	—
257	C₈	—	134	8	no	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	8	no	—
260	C₂²	C₂×Q₈×Dic₆₇	1	4	yes	—
261	C₂²	C₂×Q₈×Dic₆₇	1	2	yes	—
262	C₂²	#16864163	1	4	yes	—
263	C₂²	C₂⁴×Dic₆₇	1	1	yes	characteristic
264	C₄	—	2	4	no	—
265	C₄	—	2	4	no	—
266	C₄	—	2	16	no	—
267	C₄	C₂×Q₈×Dic₆₇	1	2	yes	—
268	C₄	C₂×Q₈×Dic₆₇	1	4	yes	—
269	C₄	C₂²×Q₈×D₆₇	1	1	yes	characteristic
270	C₄	C₂²×#11761629	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	#22569150	1	4	yes	direct product factor
273	C₂	C₂²×Q₈×Dic₆₇	1	1	yes	characteristic
274	C₂	C₂×#16864163	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂⁴×C₄
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂×#533×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂×C₄×C₆₇	C₂⁵
hypercenter	C₂×#533	D₆₇
hypocenter	C₆₇	C₂×#2961
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#533

Complements

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

N	H ≅ G/N	type
1. #22569150	C₂	direct (N × H)
2. C₂×#16863527	C₂	semidirect
11. #16863527	C₂²	semidirect
32. C₂²×C₆₇⋊C₈	Q₈	semidirect
68. C₂×C₆₇⋊C₈	C₂×Q₈	semidirect
135. C₂×C₆₇	#2961	semidirect
145. C₆₇	C₂×#2961	semidirect
272. C₂	#22569150	direct (N × H)

Maximal subgroups

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

subgroup	index
#22569150	2
C₂×#16863527	2
C₂×#16863423	2
C₂×#16863420	2
C₂×#16863352	2
C₂²×#14523954	2
C₂×#533×C₆₇	2
C₂×#2961	67

Subgroups — C2×#22569150 #31720204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22569150 #31720204