Subgroups — C₆₇×#33908 #30713159

Order 17152.

492 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	C₆₇×#3341	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2234	C₂	1	4	yes	maximal, complemented
4	C₆₇×#2226	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2215	C₂	1	2	yes	maximal, complemented
6	C₄×D₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2158	C₂	1	2	yes	maximal, complemented
9	#555×C₆₇	—	2	2	no	—
10	C₂×D₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
12	#500×C₆₇	C₂²	1	2	yes	—
13	#482×C₆₇	—	2	4	no	—
14	#476×C₆₇	—	2	2	no	—
15	#466×C₆₇	C₂²	1	4	yes	—
16	C₄×D₈×C₆₇	C₂²	1	2	yes	—
17	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
18	#427×C₆₇	C₂²	1	4	yes	—
19	#367×C₆₇	C₂²	1	4	yes	—
20	#366×C₆₇	C₂²	1	2	yes	—
21	#366×C₆₇	C₂²	1	2	yes	—
22	#366×C₆₇	C₂²	1	1	yes	characteristic
23	C₄⋊C₁₆×C₆₇	C₂²	1	1	yes	characteristic
24	#357×C₆₇	C₂²	1	2	yes	—
25	#357×C₆₇	C₂²	1	4	yes	—
26	C₂²⋊C₁₆×C₆₇	C₂²	1	2	yes	—
27	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×D₈×C₆₇	C₂³	1	2	yes	—
29	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
30	C₄⋊Q₈×C₆₇	—	2	2	no	—
31	#124×C₆₇	—	4	4	no	—
32	#121×C₆₇	—	2	4	no	—
33	#121×C₆₇	—	4	2	no	—
34	C₄×D₄×C₆₇	—	2	2	no	—
35	C₄×D₄×C₆₇	D₄	1	1	yes	characteristic
36	#117×C₆₇	—	2	4	no	—
37	#117×C₆₇	—	4	2	no	—
38	D₁₆×C₆₇	—	2	2	no	—
39	C₂×C₁₆×C₆₇	—	2	4	no	—
40	C₂×C₁₆×C₆₇	C₂³	1	2	yes	—
41	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
42	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
43	#107×C₆₇	—	2	2	no	—
44	#107×C₆₇	C₂³	1	4	yes	—
45	#107×C₆₇	C₂³	1	2	yes	—
46	#106×C₆₇	—	2	4	no	—
47	C₄⋊C₈×C₆₇	D₄	1	1	yes	characteristic
48	#102×C₆₇	—	2	4	no	—
49	#102×C₆₇	—	2	2	no	—
50	C₂²⋊C₈×C₆₇	D₄	1	2	yes	—
51	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₆₇	—	4	2	no	—
53	C₂×D₄×C₆₇	—	2	2	no	—
54	C₂×D₄×C₆₇	—	4	4	no	—
55	C₂×D₄×C₆₇	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₆₇	—	4	2	no	—
57	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
58	D₈×C₆₇	—	2	2	no	—
59	D₈×C₆₇	—	2	2	no	—
60	C₂×C₈×C₆₇	—	2	2	no	—
61	C₂×C₈×C₆₇	—	2	2	no	—
62	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
65	C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	D₈	1	1	yes	characteristic
69	C₂²⋊C₄×C₆₇	—	4	4	no	—
70	C₂²⋊C₄×C₆₇	—	4	4	no	—
71	C₂²⋊C₄×C₆₇	—	4	2	no	—
72	C₂²⋊C₄×C₆₇	D₈	1	2	yes	—
73	C₄²×C₆₇	—	4	4	no	—
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₁₆×C₆₇	—	2	2	no	—
76	C₁₆×C₆₇	—	2	1	no	—
77	C₁₆×C₆₇	#41	1	2	yes	—
78	C₂³×C₆₇	—	4	2	no	—
79	C₂³×C₆₇	C₂×D₈	1	2	yes	—
80	Q₈×C₆₇	—	16	2	no	—
81	D₄×C₆₇	—	2	2	no	—
82	D₄×C₆₇	—	4	2	no	—
83	D₄×C₆₇	—	4	2	no	—
84	D₄×C₆₇	—	16	4	no	—
85	C₂×C₄×C₆₇	—	2	2	no	—
86	C₂×C₄×C₆₇	—	2	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	4	no	—
89	C₂×C₄×C₆₇	—	8	4	no	—
90	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
91	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
94	C₈×C₆₇	—	2	2	no	—
95	C₈×C₆₇	—	2	1	no	—
96	C₈×C₆₇	C₂×#41	1	1	yes	characteristic
97	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
98	C₂²×C₆₇	—	2	2	no	—
99	C₂²×C₆₇	—	2	2	no	—
100	C₂²×C₆₇	—	2	2	no	—
101	C₂²×C₆₇	—	8	2	no	—
102	C₂²×C₆₇	—	8	2	no	—
103	C₂²×C₆₇	—	8	2	no	—
104	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
105	C₄×C₆₇	—	2	2	no	—
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	8	2	no	—
108	C₄×C₆₇	—	8	4	no	—
109	C₄×C₆₇	#508	1	2	yes	—
110	C₄×C₆₇	#547	1	1	yes	characteristic
111	C₄×C₆₇	#578	1	1	yes	characteristic
112	#33908	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂×C₆₇	—	4	2	no	—
114	C₂×C₆₇	—	16	2	no	—
115	C₂×C₆₇	#3334	1	1	yes	characteristic
116	C₂×C₆₇	C₂×#508	1	1	yes	characteristic
117	C₂×C₆₇	#3415	1	1	yes	characteristic
118	#3341	C₂×C₆₇	1	2	yes	complemented
119	#2252	C₂×C₆₇	1	1	yes	characteristic, complemented
120	#2234	C₂×C₆₇	1	4	yes	complemented
121	#2226	C₂×C₆₇	1	2	yes	complemented
122	#2215	C₂×C₆₇	1	2	yes	complemented
123	C₄×D₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
124	D₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
125	#2158	C₂×C₆₇	1	2	yes	complemented
126	C₆₇	#33908	1	1	yes	characteristic, direct product factor, Sylow
127	#555	—	2	2	no	—
128	C₂×D₁₆	C₂²×C₆₇	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
130	#500	C₂²×C₆₇	1	2	yes	—
131	#482	—	2	4	no	—
132	#476	—	2	2	no	—
133	#466	C₂²×C₆₇	1	4	yes	—
134	C₄×D₈	C₂²×C₆₇	1	2	yes	—
135	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
136	#427	C₂²×C₆₇	1	4	yes	—
137	#367	C₂²×C₆₇	1	4	yes	—
138	#366	C₂²×C₆₇	1	1	yes	characteristic
139	#366	C₂²×C₆₇	1	2	yes	—
140	#366	C₂²×C₆₇	1	2	yes	—
141	C₄⋊C₁₆	C₂²×C₆₇	1	1	yes	characteristic
142	#357	C₂²×C₆₇	1	2	yes	—
143	#357	C₂²×C₆₇	1	4	yes	—
144	C₂²⋊C₁₆	C₂²×C₆₇	1	2	yes	—
145	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
146	C₂×D₈	C₂³×C₆₇	1	2	yes	—
147	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
148	C₄⋊Q₈	—	2	2	no	—
149	#124	—	4	4	no	—
150	#121	—	2	4	no	—
151	#121	—	4	2	no	—
152	C₄×D₄	—	2	2	no	—
153	C₄×D₄	D₄×C₆₇	1	1	yes	characteristic
154	#117	—	2	4	no	—
155	#117	—	4	2	no	—
156	D₁₆	—	2	2	no	—
157	C₂×C₁₆	—	2	4	no	—
158	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
159	C₂×C₁₆	C₂³×C₆₇	1	2	yes	—
160	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
161	#107	—	2	2	no	—
162	#107	C₂³×C₆₇	1	4	yes	—
163	#107	C₂³×C₆₇	1	2	yes	—
164	#106	—	2	4	no	—
165	C₄⋊C₈	D₄×C₆₇	1	1	yes	characteristic
166	#102	—	2	2	no	—
167	#102	—	2	4	no	—
168	C₂²⋊C₈	D₄×C₆₇	1	2	yes	—
169	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
170	C₂×Q₈	—	4	2	no	—
171	C₂×D₄	—	2	2	no	—
172	C₂×D₄	—	4	4	no	—
173	C₂×D₄	D₈×C₆₇	1	1	yes	characteristic
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
176	D₈	—	2	2	no	—
177	D₈	—	2	2	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
181	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
182	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
183	C₄⋊C₄	—	2	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	D₈×C₆₇	1	1	yes	characteristic
187	C₂²⋊C₄	—	4	4	no	—
188	C₂²⋊C₄	—	4	4	no	—
189	C₂²⋊C₄	—	4	2	no	—
190	C₂²⋊C₄	D₈×C₆₇	1	2	yes	—
191	C₄²	—	4	4	no	—
192	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
193	C₁₆	—	2	2	no	—
194	C₁₆	—	2	1	no	—
195	C₁₆	#41×C₆₇	1	2	yes	—
196	C₂³	—	4	2	no	—
197	C₂³	C₂×D₈×C₆₇	1	2	yes	—
198	Q₈	—	16	2	no	—
199	D₄	—	2	2	no	—
200	D₄	—	4	2	no	—
201	D₄	—	4	2	no	—
202	D₄	—	16	4	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	4	no	—
207	C₂×C₄	—	8	4	no	—
208	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
209	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
210	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
211	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	1	no	—
214	C₈	C₂×#41×C₆₇	1	1	yes	characteristic
215	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
216	C₂²	—	2	2	no	—
217	C₂²	—	2	2	no	—
218	C₂²	—	2	2	no	—
219	C₂²	—	8	2	no	—
220	C₂²	—	8	2	no	—
221	C₂²	—	8	2	no	—
222	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	2	1	no	—
225	C₄	—	8	2	no	—
226	C₄	—	8	4	no	—
227	C₄	#508×C₆₇	1	2	yes	—
228	C₄	#547×C₆₇	1	1	yes	characteristic
229	C₄	#578×C₆₇	1	1	yes	characteristic
230	C₂	—	4	2	no	—
231	C₂	—	16	2	no	—
232	C₂	C₆₇×#3334	1	1	yes	characteristic
233	C₂	C₂×#508×C₆₇	1	1	yes	characteristic
234	C₂	C₆₇×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×D₄×C₆₇ ◁ C₈×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₆₇×#3341	C₂	semidirect
2. C₆₇×#2252	C₂	semidirect
3. C₆₇×#2234	C₂	semidirect
4. C₆₇×#2226	C₂	semidirect
5. C₆₇×#2215	C₂	semidirect
6. C₄×D₁₆×C₆₇	C₂	semidirect
7. D₄×C₁₆×C₆₇	C₂	semidirect
8. C₆₇×#2158	C₂	semidirect
112. #33908	C₆₇	direct (N × H)
118. #3341	C₂×C₆₇	semidirect
119. #2252	C₂×C₆₇	semidirect
120. #2234	C₂×C₆₇	semidirect
121. #2226	C₂×C₆₇	semidirect
122. #2215	C₂×C₆₇	semidirect
123. C₄×D₁₆	C₂×C₆₇	semidirect
124. D₄×C₁₆	C₂×C₆₇	semidirect
125. #2158	C₂×C₆₇	semidirect
126. C₆₇	#33908	direct (N × H)

Maximal subgroups

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

subgroup	index
C₆₇×#3341	2
C₆₇×#2252	2
C₆₇×#2234	2
C₆₇×#2226	2
C₆₇×#2215	2
C₄×D₁₆×C₆₇	2
D₄×C₁₆×C₆₇	2
C₆₇×#2158	2
#33908	67

Subgroups — C67×#33908 #30713159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33908 #30713159