Subgroups — #30767223

Order 17152.

394 conjugacy classes of subgroups.

The subgroup lattice has 224 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22562008	C₂	1	1	yes	maximal, characteristic
2	#22558968	C₂	1	1	yes	maximal, characteristic
3	#22558258	C₂	1	4	yes	maximal, complemented
4	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863420	—	2	1	no	—
6	C₄⋊(C₆₇⋊C₁₆)	C₂²	1	2	yes	—
7	C₂²⋊(C₆₇⋊C₁₆)	—	2	1	no	—
8	C₄×C₆₇⋊C₁₆	—	2	1	no	—
9	#16863149	—	2	1	no	—
10	#16863149	C₂²	1	2	yes	—
11	C₂²×Q₁₆×C₆₇	—	2	1	no	—
12	C₂²×Q₁₆×C₆₇	C₄	1	2	yes	—
13	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
14	#494×C₆₇	—	2	1	no	—
15	#494×C₆₇	—	2	1	no	—
16	#494×C₆₇	C₄	1	2	yes	—
17	#494×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#11761612	—	4	1	no	—
20	C₂×C₆₇⋊C₁₆	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
22	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
23	C₂²×Q₈×C₆₇	—	2	2	no	—
24	C₂×Q₁₆×C₆₇	—	2	2	no	—
25	C₂×Q₁₆×C₆₇	—	2	2	no	—
26	C₂×Q₁₆×C₆₇	—	2	2	no	—
27	C₂×Q₁₆×C₆₇	—	4	4	no	—
28	C₂×Q₁₆×C₆₇	—	4	4	no	—
29	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	—
30	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄⋊Q₈×C₆₇	—	2	2	no	—
34	C₄⋊Q₈×C₆₇	—	4	1	no	—
35	C₄⋊Q₈×C₆₇	—	4	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
37	C₂×C₄⋊C₄×C₆₇	—	4	1	no	—
38	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
39	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
43	#11761612	—	8	2	no	—
44	C₂×C₆₇⋊C₈	—	4	2	no	—
45	C₆₇⋊C₁₆	—	4	2	no	—
46	C₂×Q₈×C₆₇	—	2	4	no	—
47	C₂×Q₈×C₆₇	—	4	4	no	—
48	C₂×Q₈×C₆₇	—	4	4	no	—
49	C₂×Q₈×C₆₇	—	4	2	no	—
50	C₂²×C₄×C₆₇	—	4	2	no	—
51	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₆₇	D₈	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
54	Q₁₆×C₆₇	—	4	4	no	—
55	Q₁₆×C₆₇	—	4	4	no	—
56	Q₁₆×C₆₇	—	4	4	no	—
57	Q₁₆×C₆₇	—	4	4	no	—
58	C₂×C₈×C₆₇	—	2	2	no	—
59	C₂×C₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	4	2	no	—
65	C₄⋊C₄×C₆₇	—	4	2	no	—
66	C₄⋊C₄×C₆₇	—	4	2	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
69	C₄²×C₆₇	D₈	1	1	yes	characteristic
70	C₄²×C₆₇	SD₁₆	1	1	yes	characteristic
71	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂³×C₆₇	#102	1	1	yes	characteristic
73	Q₈×C₆₇	—	4	4	no	—
74	Q₈×C₆₇	—	4	4	no	—
75	Q₈×C₆₇	—	8	4	no	—
76	C₂×C₄×C₆₇	—	2	2	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	1	no	—
79	C₂×C₄×C₆₇	—	4	4	no	—
80	C₂×C₄×C₆₇	—	8	4	no	—
81	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
82	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
83	C₂×C₄×C₆₇	#102	1	1	yes	characteristic
84	C₂×C₄×C₆₇	SD₃₂	1	2	yes	—
85	C₂×C₄×C₆₇	Q₃₂	1	2	yes	—
86	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₈×C₆₇	—	2	1	no	—
90	C₈×C₆₇	—	2	1	no	—
91	C₈×C₆₇	#100	1	2	yes	—
92	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
93	C₆₇⋊C₈	—	8	2	no	—
94	C₂²×C₆₇	—	2	1	no	—
95	C₂²×C₆₇	—	2	1	no	—
96	C₂²×C₆₇	#358	1	2	yes	—
97	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
98	C₄×C₆₇	—	2	1	no	—
99	C₄×C₆₇	—	2	1	no	—
100	C₄×C₆₇	—	8	4	no	—
101	C₄×C₆₇	#358	1	2	yes	—
102	C₄×C₆₇	#362	1	2	yes	—
103	C₄×C₆₇	C₂×#100	1	1	yes	characteristic
104	C₄×C₆₇	#418	1	1	yes	characteristic
105	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
106	C₄×C₆₇	C₂×Q₃₂	1	1	yes	characteristic
107	#11962	—	67	1	no	maximal, Sylow
108	C₂×C₆₇	—	2	2	no	—
109	C₂×C₆₇	#1545	1	1	yes	characteristic
110	C₂×C₆₇	C₂×#358	1	1	yes	characteristic
111	C₂×C₆₇	C₂×#362	1	1	yes	characteristic
112	C₂×#494	D₆₇	1	1	yes	characteristic
113	#2162	—	67	1	no	—
114	#1567	—	67	1	no	—
115	#1362	—	67	4	no	—
116	C₆₇	#11962	1	1	yes	characteristic, complemented, Sylow
117	C₂²×Q₁₆	—	2	1	no	—
118	C₂²×Q₁₆	Dic₆₇	1	2	yes	—
119	C₂×C₄⋊Q₈	—	2	1	no	—
120	#494	—	2	1	no	—
121	#494	—	2	1	no	—
122	#494	Dic₆₇	1	2	yes	—
123	#494	C₂×D₆₇	1	2	yes	—
124	#423	—	134	1	no	—
125	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
126	C₄⋊C₁₆	—	67	2	no	—
127	C₂²⋊C₁₆	—	134	1	no	—
128	C₄×C₁₆	—	134	1	no	—
129	#335	—	67	2	no	—
130	#335	—	134	1	no	—
131	C₂²×Q₈	—	2	2	no	—
132	C₂×Q₁₆	—	2	2	no	—
133	C₂×Q₁₆	—	2	2	no	—
134	C₂×Q₁₆	—	2	2	no	—
135	C₂×Q₁₆	—	4	4	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	C₂×Dic₆₇	1	4	yes	—
138	C₂×M₁₆	—	268	1	no	—
139	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
140	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	4	1	no	—
144	C₄⋊Q₈	—	4	1	no	—
145	C₂×C₄⋊C₄	—	4	1	no	—
146	C₂×C₄⋊C₄	—	4	1	no	—
147	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
148	C₂×C₁₆	—	134	2	no	—
149	C₄⋊C₈	—	268	1	no	—
150	C₄⋊C₈	—	134	2	no	—
151	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
152	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
153	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
154	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	4	4	no	—
157	C₂×Q₈	—	4	4	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂²×C₄	—	4	2	no	—
160	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
161	C₂²×C₄	#11761613	1	1	yes	characteristic
162	C₂²×C₄	#11761616	1	1	yes	characteristic
163	Q₁₆	—	4	4	no	—
164	Q₁₆	—	4	4	no	—
165	Q₁₆	—	4	4	no	—
166	Q₁₆	—	4	4	no	—
167	M₁₆	—	536	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	268	2	no	—
171	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
173	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
174	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
175	C₄⋊C₄	—	4	2	no	—
176	C₄⋊C₄	—	4	2	no	—
177	C₄⋊C₄	—	4	2	no	—
178	C₄⋊C₄	—	4	2	no	—
179	C₄²	C₂²⋊Dic₆₇	1	1	yes	characteristic
180	C₄²	#11761613	1	1	yes	characteristic
181	C₄²	#11761616	1	1	yes	characteristic
182	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	C₁₆	—	268	2	no	—
184	C₂³	#14523964	1	1	yes	characteristic
185	Q₈	—	4	4	no	—
186	Q₈	—	4	4	no	—
187	Q₈	—	8	4	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	4	4	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	#14523964	1	1	yes	characteristic
194	C₂×C₄	#14523964	1	1	yes	characteristic
195	C₂×C₄	#14523964	1	1	yes	characteristic
196	C₂×C₄	#14523986	1	2	yes	—
197	C₂×C₄	#14523989	1	2	yes	—
198	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
199	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
200	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
201	C₈	—	2	1	no	—
202	C₈	—	2	1	no	—
203	C₈	—	536	2	no	—
204	C₈	#14523960	1	2	yes	—
205	C₈	#14524069	1	2	yes	—
206	C₂²	—	2	1	no	—
207	C₂²	—	2	1	no	—
208	C₂²	#16863200	1	2	yes	—
209	C₂²	C₂×#14523964	1	1	yes	characteristic
210	C₄	—	2	1	no	—
211	C₄	—	2	1	no	—
212	C₄	—	8	4	no	—
213	C₄	#16863200	1	2	yes	—
214	C₄	#16863212	1	2	yes	—
215	C₄	C₂×#14523960	1	1	yes	characteristic
216	C₄	#16863401	1	1	yes	characteristic
217	C₄	C₂×#14523986	1	1	yes	characteristic
218	C₄	C₂×#14523989	1	1	yes	characteristic
219	C₂	—	2	2	no	—
220	C₂	#22558877	1	1	yes	characteristic
221	C₂	C₂×#16863200	1	1	yes	characteristic
222	C₂	C₂×#16863212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523964
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂×#102
prime radical	C₂³×C₆₇	#102
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#11962
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
3. #22558258	C₂	semidirect
116. C₆₇	#11962	semidirect

Maximal subgroups

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

subgroup	index
#22562008	2
#22558968	2
#22558258	2
C₂×#494×C₆₇	2
#11962	67