Subgroups — #30767090

Order 17152.

398 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558259	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₆₈	—	2	1	no	—
6	C₂²×Dic₂₆₈	C₄	1	2	yes	—
7	C₂×#14524043	—	2	1	no	—
8	#16863776	—	2	1	no	—
9	#16863776	—	2	1	no	—
10	#16863776	C₄	1	2	yes	—
11	#16863776	C₂²	1	2	yes	—
12	#16863449	—	2	1	no	—
13	#16863149	C₂²	1	2	yes	—
14	#16863147	—	2	1	no	—
15	C₂×M₃₂×C₆₇	—	2	1	no	—
16	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₁₆×C₆₇	—	2	1	no	—
18	C₄⋊C₁₆×C₆₇	C₂²	1	2	yes	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	4	4	no	—
24	C₂×Dic₂₆₈	—	4	2	no	—
25	C₂×Dic₂₆₈	—	4	2	no	—
26	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
27	#14524043	—	2	2	no	—
28	#14524043	—	2	2	no	—
29	#14524043	—	4	1	no	—
30	#14524043	—	4	1	no	—
31	C₂×C₄⋊Dic₆₇	—	4	1	no	—
32	C₂×C₄⋊Dic₆₇	—	4	1	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
34	C₂²⋊(C₆₇⋊C₈)	—	4	1	no	—
35	C₄×C₆₇⋊C₈	—	4	1	no	—
36	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	M₃₂×C₆₇	—	4	2	no	—
40	C₂×C₁₆×C₆₇	—	2	2	no	—
41	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂×Dic₁₃₄	—	2	4	no	—
46	C₂×Dic₁₃₄	—	4	4	no	—
47	C₂×Dic₁₃₄	—	4	4	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	Dic₂₆₈	—	4	4	no	—
51	Dic₂₆₈	—	4	4	no	—
52	Dic₂₆₈	—	4	4	no	—
53	Dic₂₆₈	—	4	2	no	—
54	Dic₂₆₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄⋊Dic₆₇	—	4	2	no	—
58	C₄⋊Dic₆₇	—	4	2	no	—
59	C₄⋊Dic₆₇	—	4	2	no	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	2	no	—
64	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	D₈	1	2	yes	—
67	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
68	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
70	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₄²×C₆₇	D₈	1	1	yes	characteristic
72	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₁₆×C₆₇	—	4	2	no	—
75	C₂³×C₆₇	#102	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	#100	1	2	yes	—
79	C₂×C₄×C₆₇	#102	1	2	yes	—
80	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
85	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×Dic₆₇	—	4	4	no	—
87	C₂×Dic₆₇	—	8	4	no	—
88	Dic₁₃₄	—	4	4	no	—
89	Dic₁₃₄	—	4	4	no	—
90	Dic₁₃₄	—	8	4	no	—
91	C₈×C₆₇	—	2	1	no	—
92	C₈×C₆₇	—	2	1	no	—
93	C₈×C₆₇	#102	1	2	yes	—
94	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
95	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
96	C₆₇⋊C₈	—	8	2	no	—
97	C₂²×C₆₇	—	2	1	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	#331	1	2	yes	—
100	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
101	C₄×C₆₇	—	2	1	no	—
102	C₄×C₆₇	—	2	1	no	—
103	C₄×C₆₇	#331	1	2	yes	—
104	C₄×C₆₇	#362	1	2	yes	—
105	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
106	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
107	C₄×C₆₇	#510	1	2	yes	—
108	Dic₆₇	—	8	4	no	—
109	#11932	—	67	1	no	maximal, Sylow
110	C₂×C₆₇	—	2	2	no	—
111	C₂×C₆₇	C₂×#331	1	1	yes	characteristic
112	C₂×C₆₇	#2141	1	1	yes	characteristic
113	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
114	C₂×#494	—	67	1	no	—
115	#2150	D₆₇	1	1	yes	characteristic
116	#1564	—	67	1	no	—
117	#1362	—	67	4	no	—
118	C₆₇	#11932	1	1	yes	characteristic, complemented, Sylow
119	C₂²×Q₁₆	—	67	2	no	—
120	C₂²×Q₁₆	—	134	1	no	—
121	C₂×C₄⋊Q₈	—	134	1	no	—
122	C₂×M₃₂	—	2	1	no	—
123	#494	—	67	2	no	—
124	#494	—	67	2	no	—
125	#494	—	134	1	no	—
126	#494	—	134	1	no	—
127	#431	—	134	1	no	—
128	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
129	C₄⋊C₁₆	—	2	1	no	—
130	C₄⋊C₁₆	C₂×D₆₇	1	2	yes	—
131	#335	—	67	2	no	—
132	#334	—	134	1	no	—
133	C₂²×Q₈	—	134	2	no	—
134	C₂×Q₁₆	—	268	4	no	—
135	C₂×Q₁₆	—	268	2	no	—
136	C₂×Q₁₆	—	268	2	no	—
137	C₂×Q₁₆	—	67	4	no	—
138	C₂×Q₁₆	—	134	2	no	—
139	C₂×Q₁₆	—	134	2	no	—
140	C₂×Q₁₆	—	134	2	no	—
141	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₄⋊Q₈	—	268	1	no	—
144	C₄⋊Q₈	—	268	1	no	—
145	C₄⋊Q₈	—	134	2	no	—
146	C₄⋊Q₈	—	134	2	no	—
147	C₂×C₄⋊C₄	—	268	1	no	—
148	C₂×C₄⋊C₄	—	268	1	no	—
149	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
150	M₃₂	—	4	2	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₄⋊C₈	—	134	2	no	—
153	C₂²⋊C₈	—	268	1	no	—
154	C₄×C₈	—	268	1	no	—
155	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
156	C₄×C₈	D₂₆₈	1	1	yes	characteristic
157	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
158	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₂×Q₈	—	268	2	no	—
160	C₂×Q₈	—	268	4	no	—
161	C₂×Q₈	—	268	4	no	—
162	C₂×Q₈	—	134	4	no	—
163	C₂²×C₄	—	268	2	no	—
164	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
165	C₂²×C₄	D₅₃₆	1	1	yes	characteristic
166	C₂²×C₄	#11761617	1	1	yes	characteristic
167	Q₁₆	—	268	4	no	—
168	Q₁₆	—	268	4	no	—
169	Q₁₆	—	268	4	no	—
170	Q₁₆	—	268	2	no	—
171	Q₁₆	—	268	2	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	268	2	no	—
174	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
175	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
176	C₂×C₈	#11761613	1	2	yes	—
177	C₂×C₈	#11761615	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
180	C₄⋊C₄	—	268	2	no	—
181	C₄⋊C₄	—	268	2	no	—
182	C₄⋊C₄	—	268	2	no	—
183	C₄⋊C₄	—	268	2	no	—
184	C₄²	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₄²	D₅₃₆	1	1	yes	characteristic
186	C₄²	#11761617	1	1	yes	characteristic
187	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
188	C₁₆	—	4	2	no	—
189	C₂³	#14523965	1	1	yes	characteristic
190	Q₈	—	268	4	no	—
191	Q₈	—	268	4	no	—
192	Q₈	—	536	4	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	268	4	no	—
196	C₂×C₄	—	536	4	no	—
197	C₂×C₄	#14523959	1	2	yes	—
198	C₂×C₄	#14523963	1	2	yes	—
199	C₂×C₄	#14523965	1	1	yes	characteristic
200	C₂×C₄	#14523965	1	1	yes	characteristic
201	C₂×C₄	#14523965	1	1	yes	characteristic
202	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
204	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
205	C₈	—	2	1	no	—
206	C₈	—	2	1	no	—
207	C₈	—	536	2	no	—
208	C₈	#14523963	1	2	yes	—
209	C₈	C₂×#11761613	1	1	yes	characteristic
210	C₈	C₂×#11761615	1	1	yes	characteristic
211	C₂²	—	2	1	no	—
212	C₂²	—	2	1	no	—
213	C₂²	#16863139	1	2	yes	—
214	C₂²	C₂×#14523965	1	1	yes	characteristic
215	C₄	—	2	1	no	—
216	C₄	—	2	1	no	—
217	C₄	—	536	4	no	—
218	C₄	#16863139	1	2	yes	—
219	C₄	#16863213	1	2	yes	—
220	C₄	C₂×#14523959	1	1	yes	characteristic
221	C₄	C₂×#14523963	1	1	yes	characteristic
222	C₄	#16863839	1	2	yes	—
223	C₂	—	2	2	no	—
224	C₂	C₂×#16863139	1	1	yes	characteristic
225	C₂	#22561930	1	1	yes	characteristic
226	C₂	C₂×#16863213	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523965
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2150	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2150	D₆₇
hypocenter	C₆₇	#11932
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #22558259	C₂	semidirect
118. C₆₇	#11932	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863776	2
#22558957	2
#22558259	2
C₆₇×#2150	2
#11932	67