Subgroups — #30745801

Order 17152.

542 conjugacy classes of subgroups.

The subgroup lattice has 242 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	#22558086	C₂	1	4	yes	maximal
3	C₂×#431×C₆₇	C₂	1	1	yes	maximal, characteristic
4	#16863453	—	2	2	no	—
5	#16863453	C₄	1	4	yes	—
6	#16863449	—	2	2	no	—
7	#16863449	C₄	1	4	yes	—
8	#16863420	—	2	2	no	—
9	#16863360	C₂²	1	4	yes	—
10	C₄×#11761612	—	2	2	no	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	#431×C₆₇	—	2	2	no	—
13	#431×C₆₇	—	2	2	no	—
14	C₂×C₄⋊C₈×C₆₇	C₄	1	2	yes	—
15	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄×C₈×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)	C₂×C₄	1	8	yes	—
22	#14523954	—	2	4	no	—
23	C₄×C₆₇⋊C₈	—	2	4	no	—
24	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
25	C₂³×C₄×C₆₇	Q₈	1	1	yes	characteristic
26	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	C₄⋊C₈×C₆₇	—	2	4	no	—
37	C₂²⋊C₈×C₆₇	—	2	4	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	#11761612	—	4	8	no	—
41	C₂×C₆₇⋊C₈	—	2	8	no	—
42	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	4	no	—
44	C₂²×C₄×C₆₇	—	4	2	no	—
45	C₂²×C₄×C₆₇	—	4	2	no	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₄⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₆₇	—	2	4	no	—
59	C₂×C₈×C₆₇	—	4	4	no	—
60	C₄²×C₆₇	C₄²	1	2	yes	—
61	C₄²×C₆₇	C₄²	1	1	yes	characteristic
62	C₄²×C₆₇	C₄²	1	1	yes	characteristic
63	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
65	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂³×C₆₇	—	2	2	no	—
71	C₂³×C₆₇	—	4	1	no	—
72	C₂³×C₆₇	—	4	1	no	—
73	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	2	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	4	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	—	4	1	no	—
84	C₂×C₄×C₆₇	—	4	2	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	#95	1	2	yes	—
87	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#95	1	2	yes	—
89	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
92	C₂×C₄×C₆₇	#108	1	4	yes	—
93	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₈×C₆₇	—	4	4	no	—
97	C₆₇⋊C₈	—	4	8	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	2	2	no	—
102	C₂²×C₆₇	—	2	2	no	—
103	C₂²×C₆₇	—	4	4	no	—
104	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
105	C₂²×C₆₇	C₂×#108	1	2	yes	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	2	no	—
109	C₄×C₆₇	—	2	2	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	#338	1	2	yes	—
112	C₄×C₆₇	#341	1	2	yes	—
113	C₄×C₆₇	#341	1	2	yes	—
114	C₄×C₆₇	#410	1	2	yes	—
115	#8007	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	2	no	—
117	C₂×C₆₇	—	4	2	no	—
118	C₂×C₆₇	C₂×#338	1	1	yes	characteristic
119	C₂×C₆₇	C₂×#341	1	1	yes	characteristic
120	C₂×C₆₇	#1740	1	1	yes	characteristic
121	#2920	—	67	2	no	—
122	C₂×#431	D₆₇	1	1	yes	characteristic
123	#1293	—	67	4	no	—
124	C₆₇	#8007	1	1	yes	characteristic, complemented, Sylow
125	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
126	#432	—	67	4	no	—
127	#432	—	134	2	no	—
128	#431	—	2	2	no	—
129	#431	—	2	2	no	—
130	#431	—	67	4	no	—
131	#431	—	134	2	no	—
132	#423	—	134	2	no	—
133	C₂×C₄⋊C₈	Dic₆₇	1	2	yes	—
134	#407	—	67	4	no	—
135	C₂×C₂²⋊C₈	C₂×D₆₇	1	2	yes	—
136	C₄×M₁₆	—	134	2	no	—
137	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
138	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
139	C₂³×C₄	Dic₁₃₄	1	1	yes	characteristic
140	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
141	C₂×M₁₆	—	134	4	no	—
142	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
143	C₂×C₄²	C₄×D₆₇	1	2	yes	—
144	C₂×C₄²	C₄×D₆₇	1	2	yes	—
145	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
146	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
149	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
150	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
151	C₄⋊C₈	—	2	4	no	—
152	C₄⋊C₈	—	134	4	no	—
153	C₄⋊C₈	—	134	4	no	—
154	C₂²⋊C₈	—	2	4	no	—
155	C₂²⋊C₈	—	67	8	no	—
156	C₂²⋊C₈	—	134	4	no	—
157	#97	—	134	4	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	—	134	4	no	—
161	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
166	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
167	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
168	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
169	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
170	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
171	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
172	C₂²×C₄	C₄⋊Dic₆₇	1	1	yes	characteristic
173	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
174	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
175	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
176	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
177	M₁₆	—	268	8	no	—
178	C₂×C₈	—	2	4	no	—
179	C₂×C₈	—	4	4	no	—
180	C₂×C₈	—	134	8	no	—
181	C₄²	C₄×Dic₆₇	1	2	yes	—
182	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
183	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
184	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
185	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
186	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
187	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
188	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
189	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
190	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
191	C₂³	—	2	2	no	—
192	C₂³	—	4	1	no	—
193	C₂³	—	4	1	no	—
194	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
195	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
208	C₂×C₄	C₆₇⋊#95	1	2	yes	—
209	C₂×C₄	C₆₇⋊#95	1	2	yes	—
210	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
211	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
212	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
213	C₂×C₄	#14523979	1	4	yes	—
214	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
215	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
216	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
217	C₈	—	4	4	no	—
218	C₈	—	268	8	no	—
219	C₂²	—	2	1	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	2	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	4	no	—
225	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
226	C₂²	C₂×#14523979	1	2	yes	—
227	C₄	—	2	2	no	—
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	C₆₇⋊#338	1	2	yes	—
233	C₄	#16863161	1	2	yes	—
234	C₄	#16863161	1	2	yes	—
235	C₄	#16863370	1	2	yes	—
236	C₂	—	2	2	no	—
237	C₂	—	4	2	no	—
238	C₂	C₂×C₆₇⋊#338	1	1	yes	characteristic
239	C₂	C₂×#16863161	1	1	yes	characteristic
240	C₂	#22559942	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₆₇⋊#95
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×#431×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×#431	D₆₇
hypocenter	C₆₇	#8007
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
124. C₆₇	#8007	semidirect

Maximal subgroups

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

subgroup	index
#22568842	2
#22558086	2
C₂×#431×C₆₇	2
#8007	67