Subgroups — #30745236

Order 17152.

662 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568842	C₂	1	2	yes	maximal
2	#22558038	C₂	1	4	yes	maximal, complemented
3	C₄×#117×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863453	—	2	2	no	—
5	#16863453	—	2	2	no	—
6	#16863449	C₄	1	4	yes	complemented
7	#16863449	C₂²	1	4	yes	—
8	#16863420	—	2	2	no	—
9	#16863360	—	2	2	no	—
10	C₄×#11761612	C₄	1	4	yes	complemented
11	C₂×#117×C₆₇	C₄	1	2	yes	—
12	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
15	#376×C₆₇	C₂²	1	2	yes	—
16	C₄³×C₆₇	C₄	1	2	yes	—
17	C₂×#11761612	—	2	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
20	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
21	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
22	#14523954	—	2	4	no	—
23	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	D₄	1	2	yes	—
26	#117×C₆₇	—	2	8	no	—
27	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
29	C₂×C₄²×C₆₇	—	2	4	no	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
33	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₆₇	D₄	1	2	yes	—
37	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	Q₈	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	#95×C₆₇	—	2	4	no	—
41	#11761612	—	4	8	no	—
42	C₂×C₆₇⋊C₈	—	2	8	no	—
43	C₂⁴×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	—	2	4	no	—
45	C₂²×C₄×C₆₇	—	2	4	no	—
46	C₂²×C₄×C₆₇	—	4	4	no	—
47	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
49	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
59	C₄⋊C₄×C₆₇	—	2	8	no	—
60	C₂²⋊C₄×C₆₇	—	2	8	no	—
61	C₄²×C₆₇	—	2	4	no	—
62	C₄²×C₆₇	—	4	8	no	—
63	C₄²×C₆₇	C₄²	1	2	yes	—
64	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
70	C₄²×C₆₇	M₁₆	1	8	yes	complemented
71	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
72	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
74	C₂³×C₆₇	—	2	2	no	—
75	C₂³×C₆₇	—	4	2	no	—
76	C₂³×C₆₇	#95	1	1	yes	characteristic
77	C₂³×C₆₇	#95	1	1	yes	characteristic
78	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	4	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	2	4	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	2	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	8	no	—
92	C₂×C₄×C₆₇	#95	1	2	yes	—
93	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
95	C₂×C₄×C₆₇	#95	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
97	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
101	C₆₇⋊C₈	—	4	8	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	—	2	2	no	—
107	C₂²×C₆₇	—	4	4	no	—
108	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
109	C₂²×C₆₇	#407	1	2	yes	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	2	no	—
113	C₄×C₆₇	—	2	2	no	—
114	C₄×C₆₇	—	2	2	no	—
115	C₄×C₆₇	—	4	8	no	—
116	C₄×C₆₇	H(3,ℤ/4)	1	4	yes	—
117	C₄×C₆₇	C₄×M₁₆	1	2	yes	—
118	C₄×C₆₇	#423	1	2	yes	—
119	#7856	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	2	no	—
121	C₂×C₆₇	—	4	1	no	—
122	C₂×C₆₇	—	4	1	no	—
123	C₂×C₆₇	#1728	1	1	yes	characteristic
124	C₂×C₆₇	C₂×H(3,ℤ/4)	1	2	yes	—
125	#2920	—	67	2	no	—
126	C₄×#117	D₆₇	1	1	yes	characteristic
127	#1274	—	67	4	no	—
128	C₆₇	#7856	1	1	yes	characteristic, complemented, Sylow
129	C₂×#117	Dic₆₇	1	2	yes	—
130	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
131	#432	—	134	2	no	—
132	#432	—	134	2	no	—
133	#431	—	67	4	no	—
134	#431	—	67	4	no	—
135	#423	—	134	2	no	—
136	#407	—	134	2	no	—
137	C₄×M₁₆	—	67	4	no	—
138	C₄×C₄⋊C₄	—	2	2	no	—
139	C₄×C₂²⋊C₄	—	2	2	no	—
140	#376	C₂×D₆₇	1	2	yes	—
141	C₄³	Dic₆₇	1	2	yes	—
142	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
143	C₂³×C₄	C₂²⋊D₆₇	1	2	yes	—
144	C₂×M₁₆	—	134	4	no	—
145	#117	—	2	8	no	—
146	C₂×C₄⋊C₄	—	2	2	no	—
147	C₂×C₂²⋊C₄	—	2	2	no	—
148	C₂×C₄²	—	2	4	no	—
149	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
150	C₂×C₄²	Dic₁₃₄	1	1	yes	characteristic
151	C₂×C₄²	C₄×D₆₇	1	2	yes	—
152	C₂×C₄²	C₄×D₆₇	1	2	yes	—
153	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
154	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
155	C₂×C₄²	C₂×Dic₆₇	1	4	yes	—
156	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
157	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
158	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊C₈	—	134	4	no	—
160	C₄⋊C₈	—	134	4	no	—
161	C₂²⋊C₈	—	134	4	no	—
162	C₂²⋊C₈	—	134	4	no	—
163	#97	—	134	4	no	—
164	C₄×C₈	—	67	8	no	—
165	#95	—	2	4	no	—
166	C₂⁴	C₂²⋊Dic₆₇	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	4	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₄×Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
173	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
174	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
175	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
176	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
177	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
178	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
181	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
182	M₁₆	—	268	8	no	—
183	C₂×C₈	—	134	8	no	—
184	C₄⋊C₄	—	2	8	no	—
185	C₂²⋊C₄	—	2	8	no	—
186	C₄²	—	2	4	no	—
187	C₄²	—	4	8	no	—
188	C₄²	C₄×Dic₆₇	1	2	yes	—
189	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
190	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
191	C₄²	Dic₆₇⋊C₄	1	2	yes	—
192	C₄²	Dic₆₇⋊C₄	1	2	yes	—
193	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
194	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
195	C₄²	#11761612	1	8	yes	complemented
196	C₄²	C₂×C₄×D₆₇	1	2	yes	—
197	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
198	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
199	C₂³	—	2	2	no	—
200	C₂³	—	4	2	no	—
201	C₂³	C₆₇⋊#95	1	1	yes	characteristic
202	C₂³	C₆₇⋊#95	1	1	yes	characteristic
203	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	4	8	no	—
217	C₂×C₄	C₆₇⋊#95	1	2	yes	—
218	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
219	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
220	C₂×C₄	C₆₇⋊#95	1	2	yes	—
221	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
222	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
223	C₂×C₄	C₂×Dic₆₇⋊C₄	1	2	yes	—
224	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
225	C₂×C₄	C₂×#11761612	1	4	yes	—
226	C₈	—	268	8	no	—
227	C₂²	—	2	2	no	—
228	C₂²	—	2	1	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	2	no	—
231	C₂²	—	2	2	no	—
232	C₂²	—	4	4	no	—
233	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
234	C₂²	#16863360	1	2	yes	—
235	C₄	—	2	1	no	—
236	C₄	—	2	1	no	—
237	C₄	—	2	2	no	—
238	C₄	—	2	2	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	8	no	—
241	C₄	#16863152	1	4	yes	—
242	C₄	C₄×#11761612	1	2	yes	—
243	C₄	#16863420	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	4	1	no	—
246	C₂	—	4	1	no	—
247	C₂	#22559903	1	1	yes	characteristic
248	C₂	C₂×#16863152	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄×#117×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄×#117	D₆₇
hypocenter	C₆₇	#7856
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#117

Complements

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

N	H ≅ G/N	type
2. #22558038	C₂	semidirect
6. #16863449	C₄	semidirect
10. C₄×#11761612	C₄	semidirect
70. C₄²×C₆₇	M₁₆	semidirect
128. C₆₇	#7856	semidirect
195. C₄²	#11761612	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22558038	2
C₄×#117×C₆₇	2
#7856	67