Subgroups — C₂×C₆₇×#3342 #30740484

Order 17152.

1854 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#556×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#3342	C₂	1	16	yes	maximal, direct product factor
3	C₂×#500×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#485×C₆₇	C₂	1	4	yes	maximal, complemented
5	C₂×#475×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₂×#467×C₆₇	C₂	1	2	yes	maximal, complemented
7	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×#427×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	2	yes	—
12	#556×C₆₇	—	2	8	no	—
13	#556×C₆₇	C₂²	1	16	yes	—
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂×#126×C₆₇	—	2	4	no	—
16	C₂×#124×C₆₇	—	2	2	no	—
17	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
19	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#117×C₆₇	C₂²	1	4	yes	—
22	#500×C₆₇	C₂²	1	8	yes	complemented
23	#485×C₆₇	C₂²	1	32	yes	complemented
24	#475×C₆₇	C₂²	1	16	yes	complemented
25	#467×C₆₇	C₂²	1	16	yes	complemented
26	C₄×Q₁₆×C₆₇	C₂²	1	8	yes	complemented
27	C₈×D₄×C₆₇	C₂²	1	8	yes	—
28	#427×C₆₇	C₂²	1	16	yes	complemented
29	C₂×#107×C₆₇	C₂²	1	2	yes	—
30	C₂×#107×C₆₇	C₂²	1	2	yes	—
31	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#106×C₆₇	C₂²	1	4	yes	—
33	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₇	C₂²	1	4	yes	—
35	C₂×#103×C₆₇	C₂²	1	2	yes	—
36	C₂×C₂²⋊C₈×C₆₇	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×Q₈×C₆₇	—	2	2	no	—
39	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
40	C₂²×D₄×C₆₇	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
42	C₂×Q₁₆×C₆₇	—	2	4	no	—
43	C₂×Q₁₆×C₆₇	C₂³	1	4	yes	—
44	C₂²×C₈×C₆₇	—	2	8	no	—
45	C₂²×C₈×C₆₇	C₂³	1	8	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
47	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
49	C₄⋊Q₈×C₆₇	—	2	4	no	—
50	C₄⋊Q₈×C₆₇	C₂³	1	8	yes	—
51	#126×C₆₇	—	2	32	no	—
52	#124×C₆₇	—	2	16	no	—
53	C₂²⋊Q₈×C₆₇	—	2	8	no	—
54	C₂²⋊Q₈×C₆₇	C₂³	1	16	yes	—
55	C₄×Q₈×C₆₇	—	2	16	no	—
56	C₄×Q₈×C₆₇	—	2	4	no	—
57	C₄×Q₈×C₆₇	C₂³	1	8	yes	—
58	C₄×D₄×C₆₇	D₄	1	4	yes	—
59	C₄×D₄×C₆₇	C₂³	1	4	yes	—
60	#117×C₆₇	—	2	8	no	—
61	#117×C₆₇	C₂³	1	16	yes	—
62	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
69	C₂×C₂²⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₆₇	D₄	1	2	yes	—
71	C₂×C₄²×C₆₇	—	2	4	no	—
72	C₂×C₄²×C₆₇	—	2	2	no	—
73	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
74	#107×C₆₇	C₂³	1	8	yes	—
75	#107×C₆₇	C₂³	1	8	yes	—
76	#107×C₆₇	C₂³	1	4	yes	—
77	#106×C₆₇	C₂³	1	16	yes	—
78	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
79	#103×C₆₇	C₂³	1	16	yes	—
80	#103×C₆₇	C₂³	1	8	yes	—
81	C₂²⋊C₈×C₆₇	C₂³	1	8	yes	—
82	C₄×C₈×C₆₇	C₂³	1	4	yes	—
83	C₂⁴×C₆₇	C₂×D₄	1	2	yes	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	8	no	—
86	C₂×Q₈×C₆₇	—	2	4	no	—
87	C₂×Q₈×C₆₇	—	2	4	no	—
88	C₂×Q₈×C₆₇	—	4	8	no	—
89	C₂×Q₈×C₆₇	C₂⁴	1	4	yes	—
90	C₂×D₄×C₆₇	—	2	4	no	—
91	C₂×D₄×C₆₇	C₂×D₄	1	4	yes	—
92	C₂²×C₄×C₆₇	—	2	4	no	—
93	C₂²×C₄×C₆₇	—	2	4	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	4	no	—
97	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
98	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂⁴	1	4	yes	—
102	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₆₇	—	2	8	no	—
104	C₂×C₈×C₆₇	—	2	4	no	—
105	C₂×C₈×C₆₇	—	2	2	no	—
106	C₂×C₈×C₆₇	—	2	16	no	—
107	C₂×C₈×C₆₇	#41	1	4	yes	—
108	C₂×C₈×C₆₇	C₂⁴	1	4	yes	—
109	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
110	C₂×C₈×C₆₇	C₂⁴	1	2	yes	—
111	C₄⋊C₄×C₆₇	—	2	16	no	—
112	C₄⋊C₄×C₆₇	—	2	8	no	—
113	C₄⋊C₄×C₆₇	—	2	8	no	—
114	C₄⋊C₄×C₆₇	—	2	2	no	—
115	C₄⋊C₄×C₆₇	—	2	4	no	—
116	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
117	C₄⋊C₄×C₆₇	C₂⁴	1	4	yes	—
118	C₄⋊C₄×C₆₇	C₂⁴	1	8	yes	—
119	C₂²⋊C₄×C₆₇	—	2	16	no	—
120	C₂²⋊C₄×C₆₇	—	2	16	no	—
121	C₂²⋊C₄×C₆₇	C₂×D₄	1	8	yes	—
122	C₄²×C₆₇	—	2	16	no	—
123	C₄²×C₆₇	—	2	8	no	—
124	C₄²×C₆₇	C₂×D₄	1	2	yes	—
125	C₄²×C₆₇	C₂⁴	1	2	yes	—
126	C₂³×C₆₇	—	2	4	no	—
127	C₂³×C₆₇	—	2	4	no	—
128	C₂³×C₆₇	—	2	4	no	—
129	C₂³×C₆₇	C₂²×D₄	1	4	yes	—
130	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
131	Q₈×C₆₇	—	2	4	no	—
132	Q₈×C₆₇	—	2	4	no	—
133	Q₈×C₆₇	—	4	4	no	—
134	Q₈×C₆₇	—	4	16	no	—
135	D₄×C₆₇	—	2	8	no	—
136	C₂×C₄×C₆₇	—	2	4	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	—	2	8	no	—
139	C₂×C₄×C₆₇	—	2	4	no	—
140	C₂×C₄×C₆₇	—	2	4	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	8	no	—
144	C₂×C₄×C₆₇	—	4	4	no	—
145	C₂×C₄×C₆₇	—	4	8	no	—
146	C₂×C₄×C₆₇	—	4	4	no	—
147	C₂×C₄×C₆₇	#135	1	4	yes	—
148	C₂×C₄×C₆₇	C₂²×D₄	1	4	yes	—
149	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
152	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
153	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
154	C₂×C₄×C₆₇	C₂⁵	1	1	yes	characteristic
155	C₈×C₆₇	—	2	4	no	—
156	C₈×C₆₇	—	2	2	no	—
157	C₈×C₆₇	C₂×#41	1	4	yes	—
158	C₂²×C₆₇	—	2	4	no	—
159	C₂²×C₆₇	—	2	4	no	—
160	C₂²×C₆₇	—	2	4	no	—
161	C₂²×C₆₇	—	4	8	no	—
162	C₂²×C₆₇	#547	1	2	yes	—
163	C₂²×C₆₇	C₂×#135	1	2	yes	—
164	C₂²×C₆₇	#578	1	2	yes	—
165	C₂²×C₆₇	C₂³×D₄	1	1	yes	characteristic
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	4	4	no	—
169	C₄×C₆₇	—	4	8	no	—
170	C₄×C₆₇	—	4	4	no	—
171	C₄×C₆₇	#547	1	2	yes	—
172	C₄×C₆₇	C₂×#135	1	4	yes	—
173	C₄×C₆₇	C₂²×#41	1	1	yes	characteristic
174	C₄×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
175	C₂×#3342	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂×C₆₇	—	4	4	no	—
177	C₂×C₆₇	#3342	1	4	yes	direct product factor
178	C₂×C₆₇	C₂×#547	1	1	yes	characteristic
179	C₂×C₆₇	C₂²×#135	1	1	yes	characteristic
180	C₂×C₆₇	C₂×#578	1	1	yes	characteristic
181	C₂×#556	C₂×C₆₇	1	2	yes	—
182	#3342	C₂×C₆₇	1	16	yes	direct product factor
183	C₂×#500	C₂×C₆₇	1	1	yes	characteristic, complemented
184	C₂×#485	C₂×C₆₇	1	4	yes	complemented
185	C₂×#475	C₂×C₆₇	1	2	yes	complemented
186	C₂×#467	C₂×C₆₇	1	2	yes	complemented
187	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
188	C₂×C₈×D₄	C₂×C₆₇	1	1	yes	characteristic
189	C₂×#427	C₂×C₆₇	1	2	yes	complemented
190	C₆₇	C₂×#3342	1	1	yes	characteristic, direct product factor, Sylow
191	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
192	C₂³×C₈	C₂²×C₆₇	1	2	yes	—
193	#556	—	2	8	no	—
194	#556	C₂²×C₆₇	1	16	yes	—
195	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
196	C₂×#126	—	2	4	no	—
197	C₂×#124	—	2	2	no	—
198	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
199	C₂×C₄×Q₈	—	2	2	no	—
200	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
201	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
202	C₂×#117	C₂²×C₆₇	1	4	yes	—
203	#500	C₂²×C₆₇	1	8	yes	complemented
204	#485	C₂²×C₆₇	1	32	yes	complemented
205	#475	C₂²×C₆₇	1	16	yes	complemented
206	#467	C₂²×C₆₇	1	16	yes	complemented
207	C₄×Q₁₆	C₂²×C₆₇	1	8	yes	complemented
208	C₈×D₄	C₂²×C₆₇	1	8	yes	—
209	#427	C₂²×C₆₇	1	16	yes	complemented
210	C₂×#107	C₂²×C₆₇	1	2	yes	—
211	C₂×#107	C₂²×C₆₇	1	2	yes	—
212	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
213	C₂×#106	C₂²×C₆₇	1	4	yes	—
214	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
215	C₂×#103	C₂²×C₆₇	1	4	yes	—
216	C₂×#103	C₂²×C₆₇	1	2	yes	—
217	C₂×C₂²⋊C₈	C₂²×C₆₇	1	2	yes	—
218	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
219	C₂²×Q₈	—	2	2	no	—
220	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
221	C₂²×D₄	D₄×C₆₇	1	1	yes	characteristic
222	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
223	C₂×Q₁₆	—	2	4	no	—
224	C₂×Q₁₆	C₂³×C₆₇	1	4	yes	—
225	C₂²×C₈	—	2	8	no	—
226	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
227	C₂²×C₈	C₂³×C₆₇	1	8	yes	—
228	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
229	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
230	C₄⋊Q₈	—	2	4	no	—
231	C₄⋊Q₈	C₂³×C₆₇	1	8	yes	—
232	#126	—	2	32	no	—
233	#124	—	2	16	no	—
234	C₂²⋊Q₈	—	2	8	no	—
235	C₂²⋊Q₈	C₂³×C₆₇	1	16	yes	—
236	C₄×Q₈	—	2	16	no	—
237	C₄×Q₈	—	2	4	no	—
238	C₄×Q₈	C₂³×C₆₇	1	8	yes	—
239	C₄×D₄	D₄×C₆₇	1	4	yes	—
240	C₄×D₄	C₂³×C₆₇	1	4	yes	—
241	#117	—	2	8	no	—
242	#117	C₂³×C₆₇	1	16	yes	—
243	C₂×C₄⋊C₄	—	2	4	no	—
244	C₂×C₄⋊C₄	—	2	2	no	—
245	C₂×C₄⋊C₄	—	2	2	no	—
246	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
247	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
248	C₂×C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
249	C₂×C₂²⋊C₄	—	2	4	no	—
250	C₂×C₂²⋊C₄	—	2	4	no	—
251	C₂×C₂²⋊C₄	D₄×C₆₇	1	2	yes	—
252	C₂×C₄²	—	2	4	no	—
253	C₂×C₄²	—	2	2	no	—
254	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
255	#107	C₂³×C₆₇	1	4	yes	—
256	#107	C₂³×C₆₇	1	8	yes	—
257	#107	C₂³×C₆₇	1	8	yes	—
258	#106	C₂³×C₆₇	1	16	yes	—
259	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
260	#103	C₂³×C₆₇	1	8	yes	—
261	#103	C₂³×C₆₇	1	16	yes	—
262	C₂²⋊C₈	C₂³×C₆₇	1	8	yes	—
263	C₄×C₈	C₂³×C₆₇	1	4	yes	—
264	C₂⁴	C₂×D₄×C₆₇	1	2	yes	—
265	C₂×Q₈	—	2	2	no	—
266	C₂×Q₈	—	2	8	no	—
267	C₂×Q₈	—	2	4	no	—
268	C₂×Q₈	—	2	4	no	—
269	C₂×Q₈	—	4	8	no	—
270	C₂×Q₈	C₂⁴×C₆₇	1	4	yes	—
271	C₂×D₄	—	2	4	no	—
272	C₂×D₄	C₂×D₄×C₆₇	1	4	yes	—
273	C₂²×C₄	—	2	4	no	—
274	C₂²×C₄	—	2	4	no	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	2	4	no	—
278	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
279	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
280	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
281	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
282	C₂²×C₄	C₂⁴×C₆₇	1	4	yes	—
283	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
284	Q₁₆	—	2	8	no	—
285	C₂×C₈	—	2	4	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	2	16	no	—
288	C₂×C₈	#41×C₆₇	1	4	yes	—
289	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
290	C₂×C₈	C₂⁴×C₆₇	1	4	yes	—
291	C₂×C₈	C₂⁴×C₆₇	1	2	yes	—
292	C₄⋊C₄	—	2	16	no	—
293	C₄⋊C₄	—	2	8	no	—
294	C₄⋊C₄	—	2	8	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	4	no	—
297	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
298	C₄⋊C₄	C₂⁴×C₆₇	1	4	yes	—
299	C₄⋊C₄	C₂⁴×C₆₇	1	8	yes	—
300	C₂²⋊C₄	—	2	16	no	—
301	C₂²⋊C₄	—	2	16	no	—
302	C₂²⋊C₄	C₂×D₄×C₆₇	1	8	yes	—
303	C₄²	—	2	16	no	—
304	C₄²	—	2	8	no	—
305	C₄²	C₂×D₄×C₆₇	1	2	yes	—
306	C₄²	C₂⁴×C₆₇	1	2	yes	—
307	C₂³	—	2	4	no	—
308	C₂³	—	2	4	no	—
309	C₂³	—	2	4	no	—
310	C₂³	C₂²×D₄×C₆₇	1	4	yes	—
311	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
312	Q₈	—	2	4	no	—
313	Q₈	—	2	4	no	—
314	Q₈	—	4	4	no	—
315	Q₈	—	4	16	no	—
316	D₄	—	2	8	no	—
317	C₂×C₄	—	2	4	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	8	no	—
320	C₂×C₄	—	2	4	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	4	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	8	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	8	no	—
327	C₂×C₄	—	4	4	no	—
328	C₂×C₄	#135×C₆₇	1	4	yes	—
329	C₂×C₄	C₂²×D₄×C₆₇	1	4	yes	—
330	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
331	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
333	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
334	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
335	C₂×C₄	C₂⁵×C₆₇	1	1	yes	characteristic
336	C₈	—	2	4	no	—
337	C₈	—	2	2	no	—
338	C₈	C₂×#41×C₆₇	1	4	yes	—
339	C₂²	—	2	4	no	—
340	C₂²	—	2	4	no	—
341	C₂²	—	2	4	no	—
342	C₂²	—	4	8	no	—
343	C₂²	#547×C₆₇	1	2	yes	—
344	C₂²	C₂×#135×C₆₇	1	2	yes	—
345	C₂²	#578×C₆₇	1	2	yes	—
346	C₂²	C₂³×D₄×C₆₇	1	1	yes	characteristic
347	C₄	—	2	2	no	—
348	C₄	—	2	4	no	—
349	C₄	—	4	8	no	—
350	C₄	—	4	4	no	—
351	C₄	—	4	4	no	—
352	C₄	#547×C₆₇	1	2	yes	—
353	C₄	C₂×#135×C₆₇	1	4	yes	—
354	C₄	C₂²×#41×C₆₇	1	1	yes	characteristic
355	C₄	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
356	C₂	—	4	4	no	—
357	C₂	C₆₇×#3342	1	4	yes	direct product factor
358	C₂	C₂×#547×C₆₇	1	1	yes	characteristic
359	C₂	C₂²×#135×C₆₇	1	1	yes	characteristic
360	C₂	C₂×#578×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×D₄×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3342	C₂	direct (N × H)
3. C₂×#500×C₆₇	C₂	semidirect
4. C₂×#485×C₆₇	C₂	semidirect
5. C₂×#475×C₆₇	C₂	semidirect
6. C₂×#467×C₆₇	C₂	semidirect
7. C₂×C₄×Q₁₆×C₆₇	C₂	semidirect
9. C₂×#427×C₆₇	C₂	semidirect
22. #500×C₆₇	C₂²	semidirect
23. #485×C₆₇	C₂²	semidirect
24. #475×C₆₇	C₂²	semidirect
25. #467×C₆₇	C₂²	semidirect
26. C₄×Q₁₆×C₆₇	C₂²	semidirect
28. #427×C₆₇	C₂²	semidirect
175. C₂×#3342	C₆₇	direct (N × H)
177. C₂×C₆₇	#3342	direct (N × H)
182. #3342	C₂×C₆₇	direct (N × H)
183. C₂×#500	C₂×C₆₇	semidirect
184. C₂×#485	C₂×C₆₇	semidirect
185. C₂×#475	C₂×C₆₇	semidirect
186. C₂×#467	C₂×C₆₇	semidirect
187. C₂×C₄×Q₁₆	C₂×C₆₇	semidirect
189. C₂×#427	C₂×C₆₇	semidirect
190. C₆₇	C₂×#3342	direct (N × H)
203. #500	C₂²×C₆₇	semidirect
204. #485	C₂²×C₆₇	semidirect
205. #475	C₂²×C₆₇	semidirect
206. #467	C₂²×C₆₇	semidirect
207. C₄×Q₁₆	C₂²×C₆₇	semidirect
209. #427	C₂²×C₆₇	semidirect
357. C₂	C₆₇×#3342	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×#556×C₆₇	2
C₆₇×#3342	2
C₂×#500×C₆₇	2
C₂×#485×C₆₇	2
C₂×#475×C₆₇	2
C₂×#467×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₂×C₈×D₄×C₆₇	2
C₂×#427×C₆₇	2
C₂×#3342	67

Subgroups — C2×C67×#3342 #30740484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₆₇×#3342 #30740484