Subgroups — #30770673

Order 17152.

614 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14523983	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863351	C₂	1	1	yes	maximal, characteristic
3	#22558286	C₂	1	4	yes	maximal, complemented
4	C₂³×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11761612	C₄	1	1	yes	characteristic
6	C₂×#14524002	C₄	1	1	yes	characteristic
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	—	2	2	no	—
9	C₂×#14523983	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#16863351	—	2	2	no	—
12	#16863351	—	2	2	no	—
13	C₂×#14523953	C₄	1	2	yes	—
14	C₂×C₈×Dic₆₇	C₂²	1	2	yes	—
15	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₂²×C₁₆×C₆₇	—	2	2	no	—
17	C₂²×C₁₆×C₆₇	—	2	2	no	—
18	C₂²×C₁₆×C₆₇	C₄	1	4	yes	complemented
19	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
20	C₂×#11761612	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	C₂×#11761612	C₈	1	4	yes	—
23	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
24	#14524002	—	2	2	no	—
25	#14524002	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	C₈	1	2	yes	—
27	C₂×C₂²⋊Dic₆₇	C₈	1	2	yes	—
28	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
29	#14523983	—	2	8	no	—
30	C₂×C₆₇⋊C₁₆	—	2	2	no	—
31	C₂×C₆₇⋊C₁₆	—	2	2	no	—
32	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
33	#14523953	—	2	4	no	—
34	C₈×Dic₆₇	—	2	2	no	—
35	C₈×Dic₆₇	—	2	2	no	—
36	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₆₇	D₄	1	2	yes	—
41	C₂²×C₈×C₆₇	D₄	1	2	yes	—
42	C₂²×C₈×C₆₇	D₄	1	2	yes	—
43	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×C₁₆×C₆₇	—	2	8	no	—
46	C₂×C₁₆×C₆₇	—	2	2	no	—
47	C₂×C₁₆×C₆₇	—	2	2	no	—
48	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×Dic₆₇	C₂×C₈	1	4	yes	—
50	#11761612	—	2	8	no	—
51	C₂×C₆₇⋊C₈	—	2	2	no	—
52	C₂×C₆₇⋊C₈	—	2	2	no	—
53	C₂×C₆₇⋊C₈	C₂×C₈	1	4	yes	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₂²⋊Dic₆₇	—	2	4	no	—
56	C₄×Dic₆₇	—	2	2	no	—
57	C₄×Dic₆₇	—	2	2	no	—
58	C₆₇⋊C₁₆	—	2	4	no	—
59	C₂⁴×C₆₇	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
66	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
67	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₄²	1	2	yes	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
72	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
74	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
75	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
81	C₁₆×C₆₇	—	2	4	no	—
82	C₂³×C₆₇	—	2	4	no	—
83	C₂³×C₆₇	C₄×C₈	1	2	yes	—
84	C₂³×C₆₇	C₂²⋊C₈	1	2	yes	—
85	C₂³×C₆₇	C₄⋊C₈	1	2	yes	—
86	C₂³×C₆₇	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#95	1	4	yes	—
88	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
89	C₂×C₄×C₆₇	#97	1	4	yes	—
90	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
91	C₂×C₄×C₆₇	C₂²⋊C₈	1	4	yes	—
92	C₂×C₄×C₆₇	C₄⋊C₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₄⋊C₈	1	4	yes	—
94	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
100	C₂×Dic₆₇	—	2	4	no	—
101	C₂×Dic₆₇	—	2	8	no	—
102	C₈×C₆₇	#95	1	4	yes	—
103	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₈×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₆₇⋊C₈	—	2	4	no	—
108	C₂²×C₆₇	—	2	4	no	—
109	C₂²×C₆₇	—	2	4	no	—
110	C₂²×C₆₇	—	2	4	no	—
111	C₂²×C₆₇	#336	1	4	yes	—
112	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
113	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
114	C₂²×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₂²×C₆₇	#504	1	4	yes	—
116	C₄×C₆₇	#336	1	4	yes	—
117	C₄×C₆₇	C₂×#95	1	1	yes	characteristic
118	C₄×C₆₇	C₂×#97	1	1	yes	characteristic
119	C₄×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
121	Dic₆₇	—	4	4	no	—
122	#12545	—	67	1	no	maximal, Sylow
123	C₂×C₆₇	—	2	4	no	—
124	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
125	C₂×C₆₇	#2108	1	2	yes	—
126	C₂×C₆₇	#2113	1	2	yes	—
127	C₂×C₆₇	#2151	1	2	yes	—
128	C₂²×M₃₂	—	67	1	no	—
129	C₂³×C₁₆	D₆₇	1	1	yes	characteristic
130	C₂×#405	—	67	1	no	—
131	#1374	—	67	4	no	—
132	C₆₇	#12545	1	1	yes	characteristic, complemented, Sylow
133	C₂²×M₁₆	—	67	1	no	—
134	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
135	C₂×#117	—	67	1	no	—
136	C₂×M₃₂	—	67	4	no	—
137	C₂×M₃₂	—	134	2	no	—
138	C₂×M₃₂	—	134	2	no	—
139	C₂²×C₁₆	—	2	2	no	—
140	C₂²×C₁₆	—	2	2	no	—
141	C₂²×C₁₆	—	67	2	no	—
142	C₂²×C₁₆	Dic₆₇	1	4	yes	complemented
143	C₂²×C₁₆	C₂×D₆₇	1	2	yes	—
144	#405	—	134	2	no	—
145	#405	—	134	2	no	—
146	C₂×#97	—	67	2	no	—
147	C₂×C₄×C₈	—	67	2	no	—
148	C₂³×C₄	C₄×D₆₇	1	1	yes	characteristic
149	C₂×M₁₆	—	67	4	no	—
150	C₂×M₁₆	—	134	2	no	—
151	C₂×M₁₆	—	134	2	no	—
152	C₂²×C₈	—	67	2	no	—
153	C₂²×C₈	Dic₁₃₄	1	2	yes	—
154	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
155	C₂²×C₈	C₄×D₆₇	1	2	yes	—
156	C₂²×C₈	D₂₆₈	1	2	yes	—
157	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
158	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
159	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
160	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
161	#117	—	134	2	no	—
162	#117	—	134	2	no	—
163	C₂×C₄⋊C₄	—	67	2	no	—
164	C₂×C₂²⋊C₄	—	67	2	no	—
165	C₂×C₄²	—	67	2	no	—
166	M₃₂	—	134	8	no	—
167	C₂×C₁₆	—	2	8	no	—
168	C₂×C₁₆	—	2	2	no	—
169	C₂×C₁₆	—	2	2	no	—
170	C₂×C₁₆	—	67	4	no	—
171	C₂×C₁₆	—	134	2	no	—
172	C₂×C₁₆	—	134	2	no	—
173	C₂×C₁₆	C₂×Dic₆₇	1	4	yes	—
174	#97	—	134	4	no	—
175	C₄×C₈	—	134	2	no	—
176	C₄×C₈	—	134	2	no	—
177	C₂⁴	C₈×D₆₇	1	1	yes	characteristic
178	C₂²×C₄	—	67	4	no	—
179	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
180	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
181	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
182	C₂²×C₄	C₈×D₆₇	1	1	yes	characteristic
183	C₂²×C₄	C₈×D₆₇	1	2	yes	—
184	C₂²×C₄	#11761611	1	2	yes	—
185	C₂²×C₄	#11761611	1	2	yes	—
186	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
187	M₁₆	—	134	8	no	—
188	C₂×C₈	—	67	4	no	—
189	C₂×C₈	—	134	2	no	—
190	C₂×C₈	—	134	2	no	—
191	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
193	C₂×C₈	(C₂×D₆₇)⋊C₄	1	4	yes	—
194	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
195	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
196	C₂×C₈	Dic₆₇⋊C₄	1	4	yes	—
197	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
198	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
202	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
203	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
204	C₄⋊C₄	—	134	4	no	—
205	C₂²⋊C₄	—	134	4	no	—
206	C₄²	—	134	2	no	—
207	C₄²	—	134	2	no	—
208	C₁₆	—	2	4	no	—
209	C₁₆	—	134	4	no	—
210	C₂³	—	2	4	no	—
211	C₂³	C₈×Dic₆₇	1	2	yes	—
212	C₂³	(C₂×D₆₇)⋊C₈	1	2	yes	—
213	C₂³	Dic₆₇⋊C₈	1	2	yes	—
214	C₂³	C₂×C₈×D₆₇	1	1	yes	characteristic
215	C₂×C₄	—	134	8	no	—
216	C₂×C₄	—	134	4	no	—
217	C₂×C₄	C₆₇⋊#95	1	4	yes	—
218	C₂×C₄	C₈×Dic₆₇	1	2	yes	—
219	C₂×C₄	#14523953	1	4	yes	—
220	C₂×C₄	(C₂×D₆₇)⋊C₈	1	4	yes	—
221	C₂×C₄	(C₂×D₆₇)⋊C₈	1	2	yes	—
222	C₂×C₄	Dic₆₇⋊C₈	1	2	yes	—
223	C₂×C₄	Dic₆₇⋊C₈	1	4	yes	—
224	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
226	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
227	C₂×C₄	C₂×C₈×D₆₇	1	1	yes	characteristic
228	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
229	C₂×C₄	C₂×#11761611	1	1	yes	characteristic
230	C₈	—	134	4	no	—
231	C₈	C₆₇⋊#95	1	4	yes	—
232	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
233	C₈	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
234	C₈	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
235	C₈	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
236	C₂²	—	2	4	no	—
237	C₂²	—	2	4	no	—
238	C₂²	—	2	4	no	—
239	C₂²	#16863150	1	4	yes	—
240	C₂²	C₂×C₈×Dic₆₇	1	1	yes	characteristic
241	C₂²	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
242	C₂²	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
243	C₂²	#16863812	1	4	yes	—
244	C₄	—	268	4	no	—
245	C₄	#16863150	1	4	yes	—
246	C₄	C₂×C₆₇⋊#95	1	1	yes	characteristic
247	C₄	C₂×#14523953	1	1	yes	characteristic
248	C₄	C₂×(C₂×D₆₇)⋊C₈	1	1	yes	characteristic
249	C₄	C₂×Dic₆₇⋊C₈	1	1	yes	characteristic
250	C₂	—	2	4	no	—
251	C₂	C₂×#16863150	1	1	yes	characteristic
252	C₂	#22561805	1	2	yes	—
253	C₂	#22561822	1	2	yes	—
254	C₂	#22561967	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₆₇
commutator	C₂²×C₆₇	C₂×C₄×C₈
Fitting	C₂³×C₁₆×C₆₇	C₂
socle	C₂³×C₆₇	C₂²×C₈
prime radical	C₂⁴×C₆₇	C₂×C₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂³×C₁₆	D₆₇
hypocenter	C₆₇	#12545
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
3. #22558286	C₂	semidirect
9. C₂×#14523983	C₄	semidirect
18. C₂²×C₁₆×C₆₇	C₄	semidirect
132. C₆₇	#12545	semidirect
142. C₂²×C₁₆	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14523983	2
C₂×#16863351	2
#22558286	2
C₂³×C₁₆×C₆₇	2
#12545	67