Subgroups — C₆₇×#33238 #30712489

Order 17152.

1034 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#487×C₆₇	C₂	1	2	yes	maximal, complemented
2	C₂×#486×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#2868	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2858	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1959	C₂	1	8	yes	maximal, complemented
6	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄⋊Q₈×C₆₇	—	2	2	no	—
8	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	C₂×#127×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×#124×C₆₇	C₂²	1	4	yes	—
12	#487×C₆₇	—	2	8	no	—
13	#486×C₆₇	—	2	8	no	—
14	C₂×#103×C₆₇	C₂²	1	8	yes	—
15	C₂×#102×C₆₇	C₂²	1	8	yes	—
16	C₂×#97×C₆₇	C₂²	1	4	yes	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
18	#390×C₆₇	—	2	2	no	—
19	#390×C₆₇	—	2	2	no	—
20	#384×C₆₇	—	2	2	no	—
21	#384×C₆₇	C₂²	1	4	yes	—
22	C₄³×C₆₇	C₂²	1	1	yes	characteristic
23	C₂²×Q₈×C₆₇	—	2	2	no	—
24	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
25	C₂²×D₄×C₆₇	—	2	1	no	—
26	C₂²×D₄×C₆₇	—	2	1	no	—
27	C₂²×D₄×C₆₇	C₂³	1	2	yes	—
28	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
29	C₄⋊Q₈×C₆₇	—	2	4	no	—
30	C₄⋊Q₈×C₆₇	—	2	4	no	—
31	C₄⋊Q₈×C₆₇	—	2	4	no	—
32	C₄⋊Q₈×C₆₇	—	4	8	no	—
33	#127×C₆₇	—	2	4	no	—
34	#124×C₆₇	—	2	16	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
38	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
39	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
40	C₂×C₂²⋊C₄×C₆₇	—	2	8	no	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	D₄	1	2	yes	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
45	#103×C₆₇	—	2	16	no	—
46	#102×C₆₇	—	2	16	no	—
47	#97×C₆₇	—	2	8	no	—
48	C₄×C₈×C₆₇	D₄	1	8	yes	—
49	C₂⁴×C₆₇	—	2	2	no	—
50	C₂×Q₈×C₆₇	—	2	4	no	—
51	C₂×Q₈×C₆₇	—	2	8	no	—
52	C₂×Q₈×C₆₇	—	4	4	no	—
53	C₂×Q₈×C₆₇	—	4	8	no	—
54	C₂×D₄×C₆₇	—	2	4	no	—
55	C₂×D₄×C₆₇	—	2	8	no	—
56	C₂×D₄×C₆₇	—	4	2	no	—
57	C₂×D₄×C₆₇	—	4	2	no	—
58	C₂×D₄×C₆₇	—	4	4	no	—
59	C₂×D₄×C₆₇	—	4	4	no	—
60	C₂²×C₄×C₆₇	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	2	4	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
66	C₂×C₈×C₆₇	—	2	8	no	—
67	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₄⋊C₄×C₆₇	—	4	16	no	—
71	C₄⋊C₄×C₆₇	—	4	4	no	—
72	C₄⋊C₄×C₆₇	—	4	2	no	—
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₂²⋊C₄×C₆₇	—	4	16	no	—
75	C₄²×C₆₇	—	2	4	no	—
76	C₄²×C₆₇	—	2	4	no	—
77	C₄²×C₆₇	C₂×D₄	1	4	yes	—
78	C₄²×C₆₇	#41	1	8	yes	—
79	C₂³×C₆₇	—	4	4	no	—
80	C₂³×C₆₇	—	4	2	no	—
81	C₂³×C₆₇	—	4	4	no	—
82	C₂³×C₆₇	—	4	4	no	—
83	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
84	Q₈×C₆₇	—	4	8	no	—
85	Q₈×C₆₇	—	8	8	no	—
86	D₄×C₆₇	—	4	8	no	—
87	D₄×C₆₇	—	8	4	no	—
88	D₄×C₆₇	—	8	4	no	—
89	C₂×C₄×C₆₇	—	2	4	no	—
90	C₂×C₄×C₆₇	—	2	4	no	—
91	C₂×C₄×C₆₇	—	4	4	no	—
92	C₂×C₄×C₆₇	—	4	4	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	4	8	no	—
95	C₂×C₄×C₆₇	#124	1	8	yes	—
96	C₂×C₄×C₆₇	#127	1	2	yes	—
97	C₂×C₄×C₆₇	#127	1	2	yes	—
98	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
99	C₂×C₄×C₆₇	#137	1	4	yes	—
100	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
101	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
102	C₈×C₆₇	—	2	8	no	—
103	C₂²×C₆₇	—	8	2	no	—
104	C₂²×C₆₇	—	8	8	no	—
105	C₂²×C₆₇	—	8	2	no	—
106	C₂²×C₆₇	—	8	2	no	—
107	C₂²×C₆₇	C₂×#124	1	2	yes	—
108	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
109	C₂²×C₆₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
110	C₂²×C₆₇	C₂×#137	1	2	yes	—
111	C₄×C₆₇	—	2	8	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	—	8	4	no	—
114	C₄×C₆₇	#496	1	4	yes	—
115	C₄×C₆₇	#497	1	4	yes	—
116	C₄×C₆₇	#532	1	4	yes	—
117	#33238	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
118	C₂×C₆₇	—	16	2	no	—
119	C₂×C₆₇	#2858	1	1	yes	characteristic
120	C₂×C₆₇	#3141	1	4	yes	—
121	C₂×C₆₇	C₂×#496	1	1	yes	characteristic
122	C₂×C₆₇	C₂×#497	1	1	yes	characteristic
123	C₂×#487	C₂×C₆₇	1	2	yes	complemented
124	C₂×#486	C₂×C₆₇	1	2	yes	—
125	#2868	C₂×C₆₇	1	1	yes	characteristic, complemented
126	#2858	C₂×C₆₇	1	1	yes	characteristic
127	#1959	C₂×C₆₇	1	8	yes	complemented
128	C₄×#97	C₂×C₆₇	1	1	yes	characteristic, complemented
129	C₆₇	#33238	1	1	yes	characteristic, direct product factor, Sylow
130	C₂×C₄⋊Q₈	—	2	2	no	—
131	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
132	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
133	C₂×#127	C₂²×C₆₇	1	1	yes	characteristic
134	C₂×#124	C₂²×C₆₇	1	4	yes	—
135	#487	—	2	8	no	—
136	#486	—	2	8	no	—
137	C₂×#103	C₂²×C₆₇	1	8	yes	—
138	C₂×#102	C₂²×C₆₇	1	8	yes	—
139	C₂×#97	C₂²×C₆₇	1	4	yes	—
140	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
141	#390	—	2	2	no	—
142	#390	—	2	2	no	—
143	#384	—	2	2	no	—
144	#384	C₂²×C₆₇	1	4	yes	—
145	C₄³	C₂²×C₆₇	1	1	yes	characteristic
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
148	C₂²×D₄	—	2	1	no	—
149	C₂²×D₄	—	2	1	no	—
150	C₂²×D₄	C₂³×C₆₇	1	2	yes	—
151	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
152	C₄⋊Q₈	—	2	4	no	—
153	C₄⋊Q₈	—	2	4	no	—
154	C₄⋊Q₈	—	2	4	no	—
155	C₄⋊Q₈	—	4	8	no	—
156	#127	—	2	4	no	—
157	#124	—	2	16	no	—
158	C₂×C₄⋊C₄	—	2	8	no	—
159	C₂×C₄⋊C₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
163	C₂×C₂²⋊C₄	—	2	8	no	—
164	C₂×C₄²	D₄×C₆₇	1	2	yes	—
165	C₂×C₄²	D₄×C₆₇	1	2	yes	—
166	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
167	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
168	#103	—	2	16	no	—
169	#102	—	2	16	no	—
170	#97	—	2	8	no	—
171	C₄×C₈	D₄×C₆₇	1	8	yes	—
172	C₂⁴	—	2	2	no	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	2	8	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	4	8	no	—
177	C₂×D₄	—	2	4	no	—
178	C₂×D₄	—	2	8	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	—	4	4	no	—
182	C₂×D₄	—	4	4	no	—
183	C₂²×C₄	—	2	2	no	—
184	C₂²×C₄	—	2	4	no	—
185	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
186	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
188	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
189	C₂×C₈	—	2	8	no	—
190	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	2	4	no	—
193	C₄⋊C₄	—	4	16	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₄⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	16	no	—
198	C₄²	—	2	4	no	—
199	C₄²	—	2	4	no	—
200	C₄²	C₂×D₄×C₆₇	1	4	yes	—
201	C₄²	#41×C₆₇	1	8	yes	—
202	C₂³	—	4	4	no	—
203	C₂³	—	4	2	no	—
204	C₂³	—	4	4	no	—
205	C₂³	—	4	4	no	—
206	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
207	Q₈	—	4	8	no	—
208	Q₈	—	8	8	no	—
209	D₄	—	4	8	no	—
210	D₄	—	8	4	no	—
211	D₄	—	8	4	no	—
212	C₂×C₄	—	2	4	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	8	no	—
218	C₂×C₄	#124×C₆₇	1	8	yes	—
219	C₂×C₄	#127×C₆₇	1	2	yes	—
220	C₂×C₄	#127×C₆₇	1	2	yes	—
221	C₂×C₄	AGL(1,ℤ/8)×C₆₇	1	4	yes	—
222	C₂×C₄	#137×C₆₇	1	4	yes	—
223	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
224	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
225	C₈	—	2	8	no	—
226	C₂²	—	8	2	no	—
227	C₂²	—	8	8	no	—
228	C₂²	—	8	2	no	—
229	C₂²	—	8	2	no	—
230	C₂²	C₂×#124×C₆₇	1	2	yes	—
231	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
232	C₂²	C₂×AGL(1,ℤ/8)×C₆₇	1	2	yes	—
233	C₂²	C₂×#137×C₆₇	1	2	yes	—
234	C₄	—	2	8	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	4	no	—
237	C₄	#496×C₆₇	1	4	yes	—
238	C₄	#497×C₆₇	1	4	yes	—
239	C₄	#532×C₆₇	1	4	yes	—
240	C₂	—	16	2	no	—
241	C₂	C₆₇×#2858	1	1	yes	characteristic
242	C₂	C₆₇×#3141	1	4	yes	—
243	C₂	C₂×#496×C₆₇	1	1	yes	characteristic
244	C₂	C₂×#497×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×#127×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#487×C₆₇	C₂	semidirect
3. C₆₇×#2868	C₂	semidirect
5. C₆₇×#1959	C₂	semidirect
6. C₄×#97×C₆₇	C₂	semidirect
117. #33238	C₆₇	direct (N × H)
123. C₂×#487	C₂×C₆₇	semidirect
125. #2868	C₂×C₆₇	semidirect
127. #1959	C₂×C₆₇	semidirect
128. C₄×#97	C₂×C₆₇	semidirect
129. C₆₇	#33238	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#487×C₆₇	2
C₂×#486×C₆₇	2
C₆₇×#2868	2
C₆₇×#2858	2
C₆₇×#1959	2
C₄×#97×C₆₇	2
#33238	67

Subgroups — C67×#33238 #30712489

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33238 #30712489