Subgroups — C₂²×#16863681 #31723308

Order 17152.

2358 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863681	C₂	1	24	yes	maximal, direct product factor
3	C₂²×#14523974	C₂	1	2	yes	maximal
4	C₂²×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₂²×#14523967	C₂	1	2	yes	maximal
6	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₁₃₄	—	2	1	no	—
8	C₂³×C₆₇⋊C₈	C₂²	1	2	yes	—
9	C₂×#14524043	—	2	6	no	—
10	C₂×#14524043	C₂²	1	12	yes	—
11	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
12	C₂²×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	#16863681	C₂²	1	64	yes	direct product factor
14	C₂×#14523974	C₂²	1	24	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	12	yes	—
16	C₂×#14523967	C₂²	1	24	yes	—
17	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₆₇	—	2	6	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	12	yes	—
20	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
21	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₆₇	—	2	1	no	—
23	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×Dic₁₃₄	—	2	12	no	—
25	C₂²×Dic₁₃₄	—	4	8	no	—
26	C₂³×Dic₆₇	—	2	2	no	—
27	C₂²×C₆₇⋊C₈	—	2	8	no	—
28	C₂²×C₆₇⋊C₈	C₂³	1	12	yes	—
29	#14524043	—	2	24	no	—
30	#14524043	C₂³	1	16	yes	—
31	C₂×C₄⋊Dic₆₇	—	2	12	no	—
32	C₂×C₄⋊Dic₆₇	—	2	6	no	—
33	C₂×C₄⋊Dic₆₇	C₂³	1	12	yes	—
34	#14523974	C₂³	1	32	yes	—
35	C₄⋊(C₆₇⋊C₈)	C₂³	1	16	yes	—
36	#14523967	C₂³	1	32	yes	—
37	C₂²×Q₈×C₆₇	—	2	3	no	—
38	C₂²×Q₈×C₆₇	—	2	4	no	—
39	C₂²×Q₈×C₆₇	Q₈	1	8	yes	complemented
40	C₂²×Q₈×C₆₇	C₂³	1	6	yes	—
41	C₂³×C₄×C₆₇	—	2	1	no	—
42	C₂³×C₄×C₆₇	—	2	1	no	—
43	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
45	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₆₇	—	2	24	no	—
47	C₄×Q₈×C₆₇	C₂³	1	16	yes	—
48	C₂×C₄⋊C₄×C₆₇	—	2	12	no	—
49	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
50	C₂×C₄⋊C₄×C₆₇	C₂³	1	6	yes	—
51	C₂×C₄²×C₆₇	—	2	12	no	—
52	C₂×C₄²×C₆₇	D₄	1	6	yes	—
53	C₂×C₄²×C₆₇	C₂³	1	6	yes	—
54	C₂×Dic₁₃₄	—	2	16	no	—
55	C₂×Dic₁₃₄	—	4	48	no	—
56	C₂²×Dic₆₇	—	2	12	no	—
57	C₂²×Dic₆₇	—	4	8	no	—
58	C₂×C₆₇⋊C₈	—	2	24	no	—
59	C₂×C₆₇⋊C₈	C₂⁴	1	8	yes	—
60	C₄⋊Dic₆₇	—	2	16	no	—
61	C₄⋊Dic₆₇	—	2	12	no	—
62	C₄⋊Dic₆₇	C₂⁴	1	8	yes	—
63	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×Q₈×C₆₇	—	2	24	no	—
65	C₂×Q₈×C₆₇	—	2	6	no	—
66	C₂×Q₈×C₆₇	—	2	12	no	—
67	C₂×Q₈×C₆₇	C₂×Q₈	1	24	yes	complemented
68	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
69	C₂²×C₄×C₆₇	—	2	8	no	—
70	C₂²×C₄×C₆₇	—	2	4	no	—
71	C₂²×C₄×C₆₇	—	2	6	no	—
72	C₂²×C₄×C₆₇	—	2	6	no	—
73	C₂²×C₄×C₆₇	—	4	4	no	—
74	C₂²×C₄×C₆₇	SD₁₆	1	8	yes	—
75	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
76	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
77	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
78	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
79	C₂²×C₄×C₆₇	#41	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂⁴	1	3	yes	—
81	C₄⋊C₄×C₆₇	—	2	16	no	—
82	C₄⋊C₄×C₆₇	—	2	6	no	—
83	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
84	C₄²×C₆₇	—	2	16	no	—
85	C₄²×C₆₇	C₂×D₄	1	12	yes	—
86	C₄²×C₆₇	C₂⁴	1	4	yes	—
87	C₂³×C₆₇	C₂²⋊Q₈	1	4	yes	—
88	C₂³×C₆₇	C₂×SD₁₆	1	4	yes	—
89	C₂³×C₆₇	#137	1	4	yes	—
90	C₂³×C₆₇	C₂²×D₄	1	3	yes	—
91	Q₈×C₆₇	—	2	24	no	—
92	Q₈×C₆₇	—	2	4	no	—
93	Q₈×C₆₇	C₂²×Q₈	1	8	yes	complemented
94	C₂×C₄×C₆₇	—	2	24	no	—
95	C₂×C₄×C₆₇	—	2	12	no	—
96	C₂×C₄×C₆₇	—	2	4	no	—
97	C₂×C₄×C₆₇	—	2	4	no	—
98	C₂×C₄×C₆₇	—	4	12	no	—
99	C₂×C₄×C₆₇	C₂²⋊Q₈	1	12	yes	—
100	C₂×C₄×C₆₇	C₂×SD₁₆	1	24	yes	—
101	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
102	C₂×C₄×C₆₇	C₂²×D₄	1	3	yes	—
103	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
104	C₂×C₄×C₆₇	C₂²×Q₈	1	6	yes	—
105	C₂×C₄×C₆₇	C₂×#41	1	6	yes	—
106	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
107	C₂×Dic₆₇	—	2	8	no	—
108	C₂×Dic₆₇	—	4	24	no	—
109	Dic₁₃₄	—	4	32	no	—
110	C₆₇⋊C₈	—	2	8	no	—
111	C₂²×C₆₇	#475	1	16	yes	complemented
112	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	6	yes	—
113	C₂²×C₆₇	C₂²×SD₁₆	1	6	yes	—
114	C₂²×C₆₇	C₂×#137	1	6	yes	—
115	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
116	C₄×C₆₇	—	2	8	no	—
117	C₄×C₆₇	—	2	4	no	—
118	C₄×C₆₇	—	4	4	no	—
119	C₄×C₆₇	C₂×C₂²⋊Q₈	1	6	yes	—
120	C₄×C₆₇	C₂²×SD₁₆	1	8	yes	—
121	C₄×C₆₇	C₂³×Q₈	1	1	yes	characteristic
122	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
123	Dic₆₇	—	4	8	no	—
124	C₂²×#475	—	67	1	no	maximal, Sylow
125	C₂×C₆₇	C₂×#475	1	12	yes	complemented
126	C₂×C₆₇	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₆₇	C₂³×SD₁₆	1	1	yes	characteristic
128	C₂×C₆₇	C₂²×#137	1	1	yes	characteristic
129	C₂²×C₄⋊Q₈	—	67	1	no	—
130	C₂²×C₄×Q₈	D₆₇	1	1	yes	characteristic
131	C₂×#475	—	67	24	no	—
132	C₂²×#106	—	67	2	no	—
133	C₂²×C₄⋊C₈	—	67	1	no	—
134	C₂²×#103	—	67	2	no	—
135	C₆₇	C₂²×#475	1	1	yes	characteristic, complemented, Sylow
136	C₂³×Q₈	—	134	1	no	—
137	C₂³×Q₈	C₂×D₆₇	1	1	yes	characteristic
138	C₂³×C₈	—	67	2	no	—
139	C₂×C₄⋊Q₈	—	67	12	no	—
140	C₂×C₄⋊Q₈	—	134	6	no	—
141	C₂×C₄×Q₈	—	2	6	no	—
142	C₂×C₄×Q₈	C₂×D₆₇	1	12	yes	—
143	C₂²×C₄⋊C₄	—	2	1	no	—
144	C₂²×C₄⋊C₄	—	67	2	no	—
145	C₂²×C₄⋊C₄	—	134	1	no	—
146	C₂²×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
147	C₂²×C₄²	—	2	1	no	—
148	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
149	#475	—	67	64	no	—
150	C₂×#106	—	67	24	no	—
151	C₂×C₄⋊C₈	—	67	12	no	—
152	C₂×#103	—	67	24	no	—
153	C₂²×Q₈	—	2	3	no	—
154	C₂²×Q₈	—	2	4	no	—
155	C₂²×Q₈	—	268	8	no	—
156	C₂²×Q₈	—	134	12	no	—
157	C₂²×Q₈	Dic₁₃₄	1	8	yes	complemented
158	C₂²×Q₈	C₂²×D₆₇	1	6	yes	—
159	C₂³×C₄	—	2	1	no	—
160	C₂³×C₄	—	2	1	no	—
161	C₂³×C₄	—	134	2	no	—
162	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
163	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
164	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
165	C₂²×C₈	—	67	12	no	—
166	C₂²×C₈	—	134	8	no	—
167	C₄⋊Q₈	—	67	16	no	—
168	C₄⋊Q₈	—	134	24	no	—
169	C₄×Q₈	—	2	24	no	—
170	C₄×Q₈	C₂²×D₆₇	1	16	yes	—
171	C₂×C₄⋊C₄	—	2	12	no	—
172	C₂×C₄⋊C₄	—	2	3	no	—
173	C₂×C₄⋊C₄	—	67	12	no	—
174	C₂×C₄⋊C₄	—	134	6	no	—
175	C₂×C₄⋊C₄	—	134	12	no	—
176	C₂×C₄⋊C₄	C₂²×D₆₇	1	6	yes	—
177	C₂×C₄²	—	2	12	no	—
178	C₂×C₄²	C₂²⋊D₆₇	1	6	yes	—
179	C₂×C₄²	C₂²×D₆₇	1	6	yes	—
180	#106	—	67	32	no	—
181	C₄⋊C₈	—	67	16	no	—
182	#103	—	67	32	no	—
183	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
184	C₂×Q₈	—	2	24	no	—
185	C₂×Q₈	—	2	6	no	—
186	C₂×Q₈	—	2	12	no	—
187	C₂×Q₈	—	268	48	no	—
188	C₂×Q₈	—	134	16	no	—
189	C₂×Q₈	C₂×Dic₁₃₄	1	24	yes	complemented
190	C₂×Q₈	C₂³×D₆₇	1	4	yes	—
191	C₂²×C₄	—	2	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	6	no	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	4	4	no	—
196	C₂²×C₄	—	268	8	no	—
197	C₂²×C₄	—	134	12	no	—
198	C₂²×C₄	#11761615	1	8	yes	—
199	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
201	C₂²×C₄	C₂×C₂²⋊D₆₇	1	6	yes	—
202	C₂²×C₄	C₂×Dic₁₃₄	1	4	yes	—
203	C₂²×C₄	#11761628	1	4	yes	—
204	C₂²×C₄	C₂³×D₆₇	1	3	yes	—
205	C₂×C₈	—	67	8	no	—
206	C₂×C₈	—	134	24	no	—
207	C₄⋊C₄	—	2	16	no	—
208	C₄⋊C₄	—	2	6	no	—
209	C₄⋊C₄	—	67	8	no	—
210	C₄⋊C₄	—	134	12	no	—
211	C₄⋊C₄	—	134	16	no	—
212	C₄⋊C₄	C₂³×D₆₇	1	4	yes	—
213	C₄²	—	2	16	no	—
214	C₄²	C₂×C₂²⋊D₆₇	1	12	yes	—
215	C₄²	C₂³×D₆₇	1	4	yes	—
216	C₂³	#14524022	1	4	yes	—
217	C₂³	C₂×#11761615	1	4	yes	—
218	C₂³	#14524073	1	4	yes	—
219	C₂³	C₂²×C₂²⋊D₆₇	1	3	yes	—
220	Q₈	—	2	24	no	—
221	Q₈	—	2	4	no	—
222	Q₈	—	268	32	no	—
223	Q₈	C₂²×Dic₁₃₄	1	8	yes	complemented
224	C₂×C₄	—	2	24	no	—
225	C₂×C₄	—	2	12	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	4	no	—
228	C₂×C₄	—	4	12	no	—
229	C₂×C₄	—	268	24	no	—
230	C₂×C₄	—	134	8	no	—
231	C₂×C₄	#14524022	1	12	yes	—
232	C₂×C₄	C₂×#11761615	1	24	yes	—
233	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
234	C₂×C₄	C₂²×C₂²⋊D₆₇	1	3	yes	—
235	C₂×C₄	C₂²×C₂²⋊D₆₇	1	4	yes	—
236	C₂×C₄	C₂²×Dic₁₃₄	1	6	yes	—
237	C₂×C₄	C₂×#11761628	1	6	yes	—
238	C₂×C₄	C₂⁴×D₆₇	1	1	yes	characteristic
239	C₈	—	134	8	no	—
240	C₂²	#16863681	1	16	yes	direct product factor
241	C₂²	C₂×#14524022	1	6	yes	—
242	C₂²	C₂²×#11761615	1	6	yes	—
243	C₂²	C₂×#14524073	1	6	yes	—
244	C₂²	C₂³×C₂²⋊D₆₇	1	1	yes	characteristic
245	C₄	—	2	4	no	—
246	C₄	—	2	8	no	—
247	C₄	—	4	4	no	—
248	C₄	—	268	8	no	—
249	C₄	C₂×#14524022	1	6	yes	—
250	C₄	C₂²×#11761615	1	8	yes	—
251	C₄	C₂³×Dic₁₃₄	1	1	yes	characteristic
252	C₄	C₂²×#11761628	1	1	yes	characteristic
253	C₂	C₂×#16863681	1	12	yes	direct product factor
254	C₂	C₂²×#14524022	1	1	yes	characteristic
255	C₂	C₂³×#11761615	1	1	yes	characteristic
256	C₂	C₂²×#14524073	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂×C₂²⋊D₆₇
commutator, cosocle	C₂×C₄×C₆₇	C₂⁵
Frattini	C₂×C₄	C₂⁴×D₆₇
Fitting	C₂²×C₄×Q₈×C₆₇	C₂
socle, prime radical	C₂⁴×C₆₇	C₂×D₄
hypercenter	C₂²×C₄×Q₈	D₆₇
hypocenter	C₆₇	C₂²×#475
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#16863681	C₂	direct (N × H)
13. #16863681	C₂²	direct (N × H)
39. C₂²×Q₈×C₆₇	Q₈	semidirect
67. C₂×Q₈×C₆₇	C₂×Q₈	semidirect
93. Q₈×C₆₇	C₂²×Q₈	semidirect
111. C₂²×C₆₇	#475	semidirect
125. C₂×C₆₇	C₂×#475	semidirect
135. C₆₇	C₂²×#475	semidirect
157. C₂²×Q₈	Dic₁₃₄	semidirect
189. C₂×Q₈	C₂×Dic₁₃₄	semidirect
223. Q₈	C₂²×Dic₁₃₄	semidirect
240. C₂²	#16863681	direct (N × H)
253. C₂	C₂×#16863681	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14524043	2
C₂×#16863681	2
C₂²×#14523974	2
C₂²×C₄⋊(C₆₇⋊C₈)	2
C₂²×#14523967	2
C₂²×C₄×Q₈×C₆₇	2
C₂²×#475	67

Subgroups — C22×#16863681 #31723308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#16863681 #31723308