Subgroups — #30761639

Order 17152.

762 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863420	C₂	1	2	yes	maximal
2	C₂×C₆₇⋊#338	C₂	1	4	yes	maximal
3	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11761612	C₂²	1	4	yes	—
5	#16863420	—	2	4	no	—
6	#16863420	—	2	2	no	—
7	#16863420	—	2	2	no	—
8	C₂×C₄⋊(C₆₇⋊C₈)	C₄	1	8	yes	—
9	C₆₇⋊#338	—	2	8	no	—
10	C₂²×M₁₆×C₆₇	C₂²	1	2	yes	—
11	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
12	C₄×M₁₆×C₆₇	—	2	2	no	—
13	C₄×M₁₆×C₆₇	—	2	2	no	—
14	C₂×#97×C₆₇	C₄	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
16	C₂×#11761612	—	2	8	no	—
17	C₂×#11761612	—	2	4	no	—
18	C₂×#11761612	—	2	4	no	—
19	C₂×#11761612	—	2	8	no	—
20	C₂²×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
21	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
23	C₂³×C₄×C₆₇	Q₈	1	2	yes	—
24	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
25	C₂×M₁₆×C₆₇	—	2	4	no	—
26	C₂×M₁₆×C₆₇	—	2	4	no	—
27	C₂×M₁₆×C₆₇	D₄	1	4	yes	—
28	C₂×M₁₆×C₆₇	Q₈	1	4	yes	—
29	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₆₇	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₆₇	D₄	1	2	yes	—
34	C₂×C₄²×C₆₇	D₄	1	2	yes	—
35	C₂×C₄²×C₆₇	D₄	1	2	yes	—
36	#97×C₆₇	—	2	4	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	#11761612	—	4	8	no	—
40	#11761612	—	4	4	no	—
41	#11761612	—	4	4	no	—
42	C₂×C₆₇⋊C₈	—	2	8	no	—
43	C₂×C₆₇⋊C₈	—	2	16	no	—
44	C₂⁴×C₆₇	C₂×Q₈	1	1	yes	characteristic
45	C₂²×C₄×C₆₇	—	4	4	no	—
46	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₆₇	C₄⋊C₄	1	4	yes	—
48	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
50	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
51	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
56	C₂²×C₄×C₆₇	#41	1	4	yes	—
57	M₁₆×C₆₇	—	2	8	no	—
58	M₁₆×C₆₇	—	2	8	no	—
59	C₂×C₈×C₆₇	—	2	4	no	—
60	C₂×C₈×C₆₇	—	2	4	no	—
61	C₂×C₈×C₆₇	C₄⋊C₄	1	8	yes	—
62	C₄²×C₆₇	C₄²	1	2	yes	—
63	C₄²×C₆₇	C₄²	1	2	yes	—
64	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
65	C₄²×C₆₇	C₂²⋊C₄	1	4	yes	—
66	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
67	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
69	C₂³×C₆₇	—	4	2	no	—
70	C₂³×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂³×C₆₇	#125	1	2	yes	—
73	C₂³×C₆₇	C₄⋊Q₈	1	2	yes	—
74	C₂×C₄×C₆₇	—	2	8	no	—
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	4	4	no	—
77	C₂×C₄×C₆₇	—	4	2	no	—
78	C₂×C₄×C₆₇	#95	1	2	yes	—
79	C₂×C₄×C₆₇	#95	1	4	yes	—
80	C₂×C₄×C₆₇	#95	1	2	yes	—
81	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
85	C₂×C₄×C₆₇	#117	1	2	yes	—
86	C₂×C₄×C₆₇	C₄×D₄	1	4	yes	—
87	C₂×C₄×C₆₇	C₄×Q₈	1	4	yes	—
88	C₂×C₄×C₆₇	#121	1	4	yes	—
89	C₂×C₄×C₆₇	C₂²⋊Q₈	1	4	yes	—
90	C₂×C₄×C₆₇	#124	1	2	yes	—
91	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
93	C₈×C₆₇	—	2	8	no	—
94	C₆₇⋊C₈	—	4	8	no	—
95	C₂²×C₆₇	—	2	4	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	4	2	no	—
98	C₂²×C₆₇	—	4	2	no	—
99	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
100	C₂²×C₆₇	#382	1	1	yes	characteristic
101	C₂²×C₆₇	#384	1	1	yes	characteristic
102	C₂²×C₆₇	#389	1	4	yes	—
103	C₄×C₆₇	—	2	4	no	—
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	#338	1	4	yes	complemented
106	C₄×C₆₇	C₄×C₄⋊C₄	1	2	yes	—
107	C₄×C₆₇	#380	1	2	yes	—
108	C₄×C₆₇	#390	1	2	yes	—
109	C₄×C₆₇	#391	1	2	yes	—
110	C₄⋊#338	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	C₄⋊#95	1	1	yes	characteristic
115	C₂×C₆₇	C₂×#338	1	2	yes	—
116	C₂×#423	—	67	2	no	—
117	C₂×C₄×M₁₆	D₆₇	1	1	yes	characteristic
118	C₂×#338	—	67	4	no	—
119	C₆₇	C₄⋊#338	1	1	yes	characteristic, complemented, Sylow
120	C₂²×M₁₆	—	67	4	no	—
121	C₂²×M₁₆	C₂×D₆₇	1	2	yes	—
122	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
123	#423	—	134	4	no	—
124	#423	—	134	2	no	—
125	#423	—	134	2	no	—
126	C₂×C₄⋊C₈	—	67	8	no	—
127	C₄×M₁₆	—	2	2	no	—
128	C₄×M₁₆	—	2	2	no	—
129	C₂×#97	Dic₆₇	1	2	yes	—
130	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
131	#338	—	134	8	no	—
132	C₂³×C₄	Dic₁₃₄	1	2	yes	—
133	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
134	C₂×M₁₆	—	2	4	no	—
135	C₂×M₁₆	—	2	4	no	—
136	C₂×M₁₆	—	134	4	no	—
137	C₂×M₁₆	—	134	8	no	—
138	C₂×M₁₆	—	134	4	no	—
139	C₂×M₁₆	—	134	8	no	—
140	C₂×M₁₆	Dic₁₃₄	1	4	yes	—
141	C₂×M₁₆	D₂₆₈	1	4	yes	—
142	C₂²×C₈	—	67	8	no	—
143	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
144	C₂×C₄²	C₄×D₆₇	1	4	yes	—
145	C₂×C₄²	D₂₆₈	1	2	yes	—
146	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
147	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
148	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
149	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
150	C₄⋊C₈	—	134	8	no	—
151	C₄⋊C₈	—	134	8	no	—
152	#97	—	2	4	no	—
153	C₄×C₈	—	2	2	no	—
154	C₄×C₈	—	2	2	no	—
155	C₂⁴	C₂×Dic₁₃₄	1	1	yes	characteristic
156	C₂²×C₄	—	4	4	no	—
157	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
158	C₂²×C₄	Dic₆₇⋊C₄	1	4	yes	—
159	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
160	C₂²×C₄	C₄⋊Dic₆₇	1	2	yes	—
161	C₂²×C₄	C₂×C₄×D₆₇	1	2	yes	—
162	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
164	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
165	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
166	C₂²×C₄	C₂×Dic₁₃₄	1	2	yes	—
167	C₂²×C₄	#11761628	1	4	yes	—
168	M₁₆	—	2	8	no	—
169	M₁₆	—	2	8	no	—
170	M₁₆	—	268	8	no	—
171	M₁₆	—	268	4	no	—
172	M₁₆	—	268	4	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	134	8	no	—
176	C₂×C₈	—	134	16	no	—
177	C₂×C₈	C₄⋊Dic₆₇	1	8	yes	—
178	C₄²	C₄×Dic₆₇	1	2	yes	—
179	C₄²	C₄×Dic₆₇	1	2	yes	—
180	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
181	C₄²	(C₂×D₆₇)⋊C₄	1	4	yes	—
182	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
183	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
184	C₄²	C₂²⋊Dic₆₇	1	2	yes	—
185	C₂³	—	4	2	no	—
186	C₂³	C₂×Dic₆₇⋊C₄	1	2	yes	—
187	C₂³	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
188	C₂³	#14524035	1	2	yes	—
189	C₂³	#14524043	1	2	yes	—
190	C₂×C₄	—	2	8	no	—
191	C₂×C₄	—	2	4	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	4	2	no	—
194	C₂×C₄	C₆₇⋊#95	1	2	yes	—
195	C₂×C₄	C₆₇⋊#95	1	4	yes	—
196	C₂×C₄	C₆₇⋊#95	1	2	yes	—
197	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
198	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	2	yes	—
199	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
200	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
201	C₂×C₄	#14524002	1	2	yes	—
202	C₂×C₄	C₄×D₂₆₈	1	4	yes	—
203	C₂×C₄	C₄×Dic₁₃₄	1	4	yes	—
204	C₂×C₄	#14524017	1	4	yes	—
205	C₂×C₄	#14524022	1	4	yes	—
206	C₂×C₄	#14524031	1	2	yes	—
207	C₂×C₄	#14524040	1	1	yes	characteristic
208	C₂×C₄	#14524043	1	1	yes	characteristic
209	C₈	—	2	8	no	—
210	C₈	—	268	8	no	—
211	C₂²	—	2	4	no	—
212	C₂²	—	2	2	no	—
213	C₂²	—	4	2	no	—
214	C₂²	—	4	2	no	—
215	C₂²	C₂×C₆₇⋊#95	1	1	yes	characteristic
216	C₂²	#16863262	1	1	yes	characteristic
217	C₂²	#16863266	1	1	yes	characteristic
218	C₂²	#16863294	1	4	yes	—
219	C₄	—	2	4	no	—
220	C₄	—	2	2	no	—
221	C₄	C₆₇⋊#338	1	4	yes	complemented
222	C₄	C₄×C₄⋊Dic₆₇	1	2	yes	—
223	C₄	#16863256	1	2	yes	—
224	C₄	#16863298	1	2	yes	—
225	C₄	#16863302	1	2	yes	—
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	2	4	no	—
229	C₂	#22558439	1	1	yes	characteristic
230	C₂	C₂×C₆₇⋊#338	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₂×C₄×M₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₂×Q₈
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₂×C₄×M₁₆	D₆₇
hypocenter	C₆₇	C₄⋊#338
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
105. C₄×C₆₇	#338	semidirect
119. C₆₇	C₄⋊#338	semidirect
221. C₄	C₆₇⋊#338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863420	2
C₂×C₆₇⋊#338	2
C₂×C₄×M₁₆×C₆₇	2
C₄⋊#338	67