Subgroups — C₆₇×#7918 #30687169

Order 17152.

666 conjugacy classes of subgroups.

The subgroup lattice has 236 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#431×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1281	C₂	1	4	yes	maximal, complemented
4	C₂²×M₁₆×C₆₇	C₄	1	2	yes	—
5	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
6	#431×C₆₇	—	2	2	no	—
7	#431×C₆₇	—	2	2	no	—
8	#431×C₆₇	C₄	1	4	yes	—
9	#431×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄⋊C₈×C₆₇	—	2	2	no	—
11	C₂×C₂²⋊C₈×C₆₇	C₄	1	4	yes	—
12	C₄×M₁₆×C₆₇	—	2	2	no	—
13	C₄×M₁₆×C₆₇	—	2	2	no	—
14	C₂×#97×C₆₇	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₆₇	—	2	2	no	—
16	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
17	C₂³×C₄×C₆₇	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
19	C₂×M₁₆×C₆₇	—	2	4	no	—
20	C₂×M₁₆×C₆₇	—	2	4	no	—
21	C₂×M₁₆×C₆₇	C₈	1	8	yes	complemented
22	C₂²×C₈×C₆₇	—	2	4	no	—
23	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄²×C₆₇	D₄	1	2	yes	—
27	C₂×C₄²×C₆₇	D₄	1	2	yes	—
28	C₂×C₄²×C₆₇	D₄	1	2	yes	—
29	C₂×C₄²×C₆₇	Q₈	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
31	C₄⋊C₈×C₆₇	—	2	4	no	—
32	C₄⋊C₈×C₆₇	—	2	4	no	—
33	C₂²⋊C₈×C₆₇	—	2	4	no	—
34	C₂²⋊C₈×C₆₇	C₂×C₄	1	8	yes	—
35	#97×C₆₇	—	2	4	no	—
36	C₄×C₈×C₆₇	—	2	4	no	—
37	C₄×C₈×C₆₇	—	2	2	no	—
38	C₄×C₈×C₆₇	—	2	2	no	—
39	C₄×C₈×C₆₇	—	2	4	no	—
40	C₂⁴×C₆₇	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₆₇	—	4	4	no	—
42	C₂²×C₄×C₆₇	C₄²	1	2	yes	—
43	C₂²×C₄×C₆₇	C₄²	1	1	yes	characteristic
44	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
45	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₂²×C₄×C₆₇	C₂²⋊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	C₂×C₈	1	4	yes	—
51	C₂²×C₄×C₆₇	M₁₆	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
54	M₁₆×C₆₇	—	2	16	no	—
55	C₂×C₈×C₆₇	—	2	4	no	—
56	C₂×C₈×C₆₇	—	2	4	no	—
57	C₂×C₈×C₆₇	—	2	8	no	—
58	C₂×C₈×C₆₇	—	4	8	no	—
59	C₂×C₈×C₆₇	C₂×C₈	1	8	yes	complemented
60	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
62	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₄²×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
64	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
65	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
66	C₄²×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₄²×C₆₇	C₂×D₄	1	2	yes	—
69	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
70	C₂³×C₆₇	—	4	2	no	—
71	C₂³×C₆₇	#95	1	2	yes	—
72	C₂³×C₆₇	C₄×C₈	1	2	yes	—
73	C₂³×C₆₇	#97	1	2	yes	—
74	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
75	C₂×C₄×C₆₇	—	2	4	no	—
76	C₂×C₄×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	4	2	no	—
79	C₂×C₄×C₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	4	2	no	—
81	C₂×C₄×C₆₇	#95	1	2	yes	—
82	C₂×C₄×C₆₇	#95	1	2	yes	—
83	C₂×C₄×C₆₇	#95	1	2	yes	—
84	C₂×C₄×C₆₇	C₄×C₈	1	2	yes	—
85	C₂×C₄×C₆₇	#97	1	2	yes	—
86	C₂×C₄×C₆₇	C₂²⋊C₈	1	8	yes	—
87	C₂×C₄×C₆₇	C₄⋊C₈	1	8	yes	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
89	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
93	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
94	C₈×C₆₇	—	2	8	no	—
95	C₈×C₆₇	—	4	8	no	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	4	2	no	—
100	C₂²×C₆₇	—	4	2	no	—
101	C₂²×C₆₇	#336	1	4	yes	—
102	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×C₄×C₈	1	1	yes	characteristic
104	C₂²×C₆₇	C₂×#97	1	1	yes	characteristic
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	2	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	#336	1	4	yes	—
109	C₄×C₆₇	#344	1	4	yes	—
110	C₄×C₆₇	C₂×C₂²⋊C₈	1	2	yes	—
111	C₄×C₆₇	C₂×C₄⋊C₈	1	2	yes	—
112	#7918	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	2	4	no	—
114	C₂×C₆₇	—	2	1	no	—
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	C₂×#336	1	1	yes	characteristic
117	C₂×C₆₇	C₂×#344	1	2	yes	—
118	C₂×#431	C₂×C₆₇	1	2	yes	—
119	C₂×C₄×M₁₆	C₂×C₆₇	1	1	yes	characteristic
120	#1281	C₂×C₆₇	1	4	yes	complemented
121	C₆₇	#7918	1	1	yes	characteristic, direct product factor, Sylow
122	C₂²×M₁₆	C₄×C₆₇	1	2	yes	—
123	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
124	#431	—	2	2	no	—
125	#431	—	2	2	no	—
126	#431	C₄×C₆₇	1	4	yes	—
127	#431	C₂²×C₆₇	1	4	yes	—
128	C₂×C₄⋊C₈	—	2	2	no	—
129	C₂×C₂²⋊C₈	C₄×C₆₇	1	4	yes	—
130	C₄×M₁₆	—	2	2	no	—
131	C₄×M₁₆	—	2	2	no	—
132	C₂×#97	C₂²×C₆₇	1	2	yes	—
133	C₂×C₄×C₈	—	2	2	no	—
134	C₂×C₄×C₈	C₄×C₆₇	1	2	yes	—
135	C₂³×C₄	C₂×C₄×C₆₇	1	2	yes	—
136	C₂³×C₄	C₂×C₄×C₆₇	1	1	yes	characteristic
137	C₂×M₁₆	—	2	4	no	—
138	C₂×M₁₆	—	2	4	no	—
139	C₂×M₁₆	C₈×C₆₇	1	8	yes	complemented
140	C₂²×C₈	—	2	4	no	—
141	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
142	C₂×C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
143	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
144	C₂×C₄²	D₄×C₆₇	1	2	yes	—
145	C₂×C₄²	D₄×C₆₇	1	2	yes	—
146	C₂×C₄²	D₄×C₆₇	1	2	yes	—
147	C₂×C₄²	Q₈×C₆₇	1	2	yes	—
148	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
149	C₄⋊C₈	—	2	4	no	—
150	C₄⋊C₈	—	2	4	no	—
151	C₂²⋊C₈	—	2	4	no	—
152	C₂²⋊C₈	C₂×C₄×C₆₇	1	8	yes	—
153	#97	—	2	4	no	—
154	C₄×C₈	—	2	4	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	4	no	—
158	C₂⁴	C₄²×C₆₇	1	1	yes	characteristic
159	C₂²×C₄	—	4	4	no	—
160	C₂²×C₄	C₄²×C₆₇	1	2	yes	—
161	C₂²×C₄	C₄²×C₆₇	1	1	yes	characteristic
162	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
163	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
164	C₂²×C₄	C₂²⋊C₄×C₆₇	1	2	yes	—
165	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
166	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
167	C₂²×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
168	C₂²×C₄	C₂×C₈×C₆₇	1	4	yes	—
169	C₂²×C₄	M₁₆×C₆₇	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
172	M₁₆	—	2	16	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	2	8	no	—
176	C₂×C₈	—	4	8	no	—
177	C₂×C₈	C₂×C₈×C₆₇	1	8	yes	complemented
178	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
179	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
180	C₄²	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
181	C₄²	C₂²⋊C₄×C₆₇	1	2	yes	—
182	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
183	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
184	C₄²	C₄⋊C₄×C₆₇	1	2	yes	—
185	C₄²	C₂×D₄×C₆₇	1	2	yes	—
186	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
187	C₄²	C₂×Q₈×C₆₇	1	1	yes	characteristic
188	C₂³	—	4	2	no	—
189	C₂³	#95×C₆₇	1	2	yes	—
190	C₂³	C₄×C₈×C₆₇	1	2	yes	—
191	C₂³	#97×C₆₇	1	2	yes	—
192	C₂³	C₂×C₄²×C₆₇	1	1	yes	characteristic
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	4	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	#95×C₆₇	1	2	yes	—
200	C₂×C₄	#95×C₆₇	1	2	yes	—
201	C₂×C₄	#95×C₆₇	1	2	yes	—
202	C₂×C₄	C₄×C₈×C₆₇	1	2	yes	—
203	C₂×C₄	#97×C₆₇	1	2	yes	—
204	C₂×C₄	C₂²⋊C₈×C₆₇	1	8	yes	—
205	C₂×C₄	C₄⋊C₈×C₆₇	1	8	yes	—
206	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
207	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	2	yes	—
209	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂²×C₈×C₆₇	1	2	yes	—
211	C₂×C₄	C₂×M₁₆×C₆₇	1	2	yes	—
212	C₈	—	2	8	no	—
213	C₈	—	4	8	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	2	no	—
218	C₂²	—	4	2	no	—
219	C₂²	#336×C₆₇	1	4	yes	—
220	C₂²	C₂×#95×C₆₇	1	1	yes	characteristic
221	C₂²	C₂×C₄×C₈×C₆₇	1	1	yes	characteristic
222	C₂²	C₂×#97×C₆₇	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	#336×C₆₇	1	4	yes	—
227	C₄	#344×C₆₇	1	4	yes	—
228	C₄	C₂×C₂²⋊C₈×C₆₇	1	2	yes	—
229	C₄	C₂×C₄⋊C₈×C₆₇	1	2	yes	—
230	C₂	—	2	4	no	—
231	C₂	—	2	1	no	—
232	C₂	—	2	1	no	—
233	C₂	C₂×#336×C₆₇	1	1	yes	characteristic
234	C₂	C₂×#344×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₄²×C₆₇	C₂²⋊C₄
commutator	C₂²	C₂×C₄×C₈×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#95
prime radical	C₂⁴×C₆₇	C₄²
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1281	C₂	semidirect
21. C₂×M₁₆×C₆₇	C₈	semidirect
59. C₂×C₈×C₆₇	C₂×C₈	semidirect
112. #7918	C₆₇	direct (N × H)
120. #1281	C₂×C₆₇	semidirect
121. C₆₇	#7918	direct (N × H)
139. C₂×M₁₆	C₈×C₆₇	semidirect
177. C₂×C₈	C₂×C₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#431×C₆₇	2
C₂×C₄×M₁₆×C₆₇	2
C₆₇×#1281	2
#7918	67

Subgroups — C67×#7918 #30687169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#7918 #30687169