Subgroups — #30770901

Order 17152.

462 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863178	C₂	1	2	yes	maximal
2	C₂×#16863174	C₂	1	2	yes	maximal, complemented
3	#22558291	C₂	1	2	yes	maximal, complemented
4	C₂×#405×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523983	C₂²	1	4	yes	—
6	#16863178	C₄	1	8	yes	complemented
7	#16863174	—	2	4	no	—
8	C₂²×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂³×C₈×C₆₇	C₄	1	1	yes	characteristic
10	C₂×#117×C₆₇	C₄	1	1	yes	characteristic
11	#405×C₆₇	—	2	4	no	—
12	C₂×#97×C₆₇	C₄	1	1	yes	characteristic
13	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
16	#14523983	—	2	4	no	—
17	#14523983	C₂×C₄	1	8	yes	—
18	C₂×C₆₇⋊C₁₆	—	2	4	no	—
19	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
20	C₂×M₁₆×C₆₇	—	2	2	no	—
21	C₂×M₁₆×C₆₇	—	2	2	no	—
22	C₂×M₁₆×C₆₇	C₂×C₄	1	4	yes	—
23	C₂²×C₈×C₆₇	—	2	2	no	—
24	C₂²×C₈×C₆₇	—	2	2	no	—
25	C₂²×C₈×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	#117×C₆₇	—	2	2	no	—
31	#117×C₆₇	—	2	2	no	—
32	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
33	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
34	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
36	#97×C₆₇	—	2	2	no	—
37	#97×C₆₇	D₄	1	4	yes	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	2	no	—
40	C₆₇⋊C₁₆	—	2	8	no	—
41	C₂⁴×C₆₇	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₆₇	—	2	1	no	—
43	C₂²×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₄×C₆₇	—	2	1	no	—
45	C₂²×C₄×C₆₇	—	2	1	no	—
46	C₂²×C₄×C₆₇	—	2	1	no	—
47	C₂²×C₄×C₆₇	—	2	1	no	—
48	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
49	C₂²×C₄×C₆₇	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₆₇	M₁₆	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	M₁₆×C₆₇	—	2	8	no	—
53	C₂×C₈×C₆₇	—	2	8	no	—
54	C₂×C₈×C₆₇	—	2	1	no	—
55	C₂×C₈×C₆₇	—	2	1	no	—
56	C₂×C₈×C₆₇	—	2	1	no	—
57	C₂×C₈×C₆₇	—	2	1	no	—
58	C₂×C₈×C₆₇	—	2	1	no	—
59	C₂×C₈×C₆₇	—	2	1	no	—
60	C₂×C₈×C₆₇	C₄²	1	2	yes	—
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
62	C₂×C₈×C₆₇	M₁₆	1	2	yes	—
63	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
64	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
65	C₂×C₈×C₆₇	#41	1	2	yes	—
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₄²×C₆₇	—	2	1	no	—
69	C₄²×C₆₇	—	2	1	no	—
70	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
71	C₄²×C₆₇	M₁₆	1	2	yes	—
72	C₂³×C₆₇	—	2	1	no	—
73	C₂³×C₆₇	—	2	1	no	—
74	C₂³×C₆₇	—	4	2	no	—
75	C₂³×C₆₇	#97	1	2	yes	—
76	C₂³×C₆₇	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	4	2	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	#97	1	2	yes	—
88	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
94	C₈×C₆₇	—	2	2	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₈×C₆₇	—	2	2	no	—
97	C₈×C₆₇	C₄×D₄	1	4	yes	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	4	2	no	—
101	C₂²×C₆₇	—	4	2	no	—
102	C₂²×C₆₇	—	4	2	no	—
103	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
104	C₂²×C₆₇	#407	1	1	yes	characteristic
105	C₂²×C₆₇	#432	1	1	yes	characteristic
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	4	2	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
110	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
111	C₄×C₆₇	#407	1	1	yes	characteristic
112	C₄×C₆₇	#432	1	1	yes	characteristic
113	#12608	—	67	1	no	maximal, Sylow
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	—	4	2	no	—
117	C₂×C₆₇	#1749	1	1	yes	characteristic
118	C₂×C₆₇	#2166	1	2	yes	—
119	C₂×#405	D₆₇	1	1	yes	characteristic
120	C₂×#349	—	67	2	no	—
121	C₂×#347	—	67	2	no	—
122	#1376	—	67	2	no	—
123	C₆₇	#12608	1	1	yes	characteristic, complemented, Sylow
124	C₂²×M₁₆	C₂×D₆₇	1	1	yes	characteristic
125	C₂³×C₈	Dic₆₇	1	1	yes	characteristic
126	C₂×#117	Dic₆₇	1	1	yes	characteristic
127	C₂×M₃₂	—	67	4	no	—
128	#405	—	2	4	no	—
129	C₂×#97	Dic₆₇	1	1	yes	characteristic
130	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
131	C₂×C₄×C₈	Dic₆₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	#349	—	67	8	no	—
134	#347	—	134	4	no	—
135	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
136	C₂×M₁₆	—	2	2	no	—
137	C₂×M₁₆	—	2	2	no	—
138	C₂×M₁₆	C₄×D₆₇	1	4	yes	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
143	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
145	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
146	#117	—	2	2	no	—
147	#117	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₂²⋊C₄	—	2	1	no	—
150	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
151	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
152	M₃₂	—	67	8	no	—
153	M₃₂	—	134	4	no	—
154	C₂×C₁₆	—	134	4	no	—
155	#97	—	2	2	no	—
156	#97	C₂²⋊D₆₇	1	4	yes	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₂⁴	#11761612	1	1	yes	characteristic
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	C₄×Dic₆₇	1	2	yes	—
167	C₂²×C₄	#11761612	1	1	yes	characteristic
168	C₂²×C₄	#11761612	1	2	yes	—
169	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
170	M₁₆	—	2	8	no	—
171	C₂×C₈	—	2	8	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	1	no	—
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
179	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
180	C₂×C₈	#11761612	1	2	yes	—
181	C₂×C₈	C₂×C₄×D₆₇	1	2	yes	—
182	C₂×C₈	C₂×C₂²⋊D₆₇	1	2	yes	—
183	C₂×C₈	#11761628	1	2	yes	—
184	C₄⋊C₄	—	2	4	no	—
185	C₂²⋊C₄	—	2	4	no	—
186	C₄²	—	2	1	no	—
187	C₄²	—	2	1	no	—
188	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
189	C₄²	#11761612	1	2	yes	—
190	C₁₆	—	134	8	no	—
191	C₂³	—	2	1	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	4	2	no	—
194	C₂³	#14523954	1	2	yes	—
195	C₂³	C₂×#11761612	1	1	yes	characteristic
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	#14523954	1	2	yes	—
207	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
209	C₂×C₄	#14524002	1	1	yes	characteristic
210	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
211	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
212	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
213	C₈	—	2	2	no	—
214	C₈	—	2	2	no	—
215	C₈	—	2	2	no	—
216	C₈	C₄×C₂²⋊D₆₇	1	4	yes	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	4	2	no	—
220	C₂²	—	4	2	no	—
221	C₂²	—	4	2	no	—
222	C₂²	C₂×#14523954	1	1	yes	characteristic
223	C₂²	#16863360	1	1	yes	characteristic
224	C₂²	#16863453	1	1	yes	characteristic
225	C₄	—	2	2	no	—
226	C₄	—	4	2	no	—
227	C₄	—	4	2	no	—
228	C₄	C₄×C₂²⋊Dic₆₇	1	1	yes	characteristic
229	C₄	C₄×#11761612	1	1	yes	characteristic
230	C₄	#16863360	1	1	yes	characteristic
231	C₄	#16863453	1	1	yes	characteristic
232	C₂	—	2	1	no	—
233	C₂	—	2	1	no	—
234	C₂	—	4	2	no	—
235	C₂	#22559971	1	1	yes	characteristic
236	C₂	#22562025	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14524002
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#405×C₆₇	C₂
socle	C₂²×C₆₇	#432
prime radical	C₂⁴×C₆₇	M₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#405	D₆₇
hypocenter	C₆₇	#12608
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. C₂×#16863174	C₂	semidirect
3. #22558291	C₂	semidirect
6. #16863178	C₄	semidirect
123. C₆₇	#12608	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863178	2
C₂×#16863174	2
#22558291	2
C₂×#405×C₆₇	2
#12608	67