Subgroups — C₆₇×#22182 #30701433

Order 17152.

894 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#467×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Q₈×C₆₇	C₂	1	2	yes	maximal
3	C₆₇×#1928	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1921	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1892	C₂	1	2	yes	maximal
6	C₆₇×#1878	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1808	C₂	1	1	yes	maximal, characteristic
9	C₂×#336×C₆₇	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#467×C₆₇	C₄	1	8	yes	complemented
17	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×#107×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#103×C₆₇	C₂²	1	2	yes	—
21	C₂×#103×C₆₇	C₂²	1	2	yes	—
22	C₂×#103×C₆₇	C₂²	1	2	yes	—
23	#391×C₆₇	C₂²	1	2	yes	—
24	#389×C₆₇	C₂²	1	2	yes	—
25	#387×C₆₇	—	2	2	no	—
26	#385×C₆₇	—	2	2	no	—
27	#380×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄×C₂²⋊C₄×C₆₇	—	2	2	no	—
31	#340×C₆₇	C₂²	1	2	yes	—
32	#340×C₆₇	C₂²	1	2	yes	—
33	#340×C₆₇	C₂²	1	2	yes	—
34	#336×C₆₇	C₂²	1	2	yes	—
35	#336×C₆₇	C₂²	1	2	yes	—
36	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×Q₁₆×C₆₇	—	4	1	no	—
41	C₂×Q₁₆×C₆₇	—	4	1	no	—
42	C₂×Q₁₆×C₆₇	C₂×C₄	1	4	yes	complemented
43	C₂²×C₈×C₆₇	—	2	2	no	—
44	C₂²×C₈×C₆₇	—	4	1	no	—
45	C₂²×C₈×C₆₇	—	4	1	no	—
46	C₂²×C₈×C₆₇	—	4	1	no	—
47	C₂²×C₈×C₆₇	—	4	1	no	—
48	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₂²⋊Q₈×C₆₇	—	2	4	no	—
53	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
54	C₄×Q₈×C₆₇	—	4	2	no	—
55	C₄×Q₈×C₆₇	—	4	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
59	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
64	C₂×C₄²×C₆₇	—	2	2	no	—
65	C₂×C₄²×C₆₇	—	2	2	no	—
66	C₂×C₄²×C₆₇	—	4	1	no	—
67	C₂×C₄²×C₆₇	—	4	1	no	—
68	C₂×C₄²×C₆₇	D₄	1	2	yes	—
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	#107×C₆₇	—	4	1	no	—
71	#107×C₆₇	—	4	1	no	—
72	#107×C₆₇	C₂×C₄	1	4	yes	complemented
73	#103×C₆₇	—	4	2	no	—
74	#103×C₆₇	—	4	2	no	—
75	#103×C₆₇	C₂×C₄	1	8	yes	complemented
76	#95×C₆₇	—	2	2	no	—
77	#95×C₆₇	—	2	2	no	—
78	#95×C₆₇	—	2	2	no	—
79	#95×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
82	C₂×Q₈×C₆₇	—	2	2	no	—
83	C₂×Q₈×C₆₇	—	2	2	no	—
84	C₂×Q₈×C₆₇	—	2	2	no	—
85	C₂×Q₈×C₆₇	—	2	2	no	—
86	C₂×Q₈×C₆₇	—	2	2	no	—
87	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
88	C₂²×C₄×C₆₇	—	2	1	no	—
89	C₂²×C₄×C₆₇	—	2	1	no	—
90	C₂²×C₄×C₆₇	—	2	1	no	—
91	C₂²×C₄×C₆₇	—	2	1	no	—
92	C₂²×C₄×C₆₇	—	2	2	no	—
93	C₂²×C₄×C₆₇	—	2	2	no	—
94	C₂²×C₄×C₆₇	—	2	2	no	—
95	C₂²×C₄×C₆₇	—	2	2	no	—
96	C₂²×C₄×C₆₇	—	2	2	no	—
97	C₂²×C₄×C₆₇	—	4	1	no	—
98	C₂²×C₄×C₆₇	—	4	1	no	—
99	C₂²×C₄×C₆₇	—	4	1	no	—
100	C₂²×C₄×C₆₇	—	4	1	no	—
101	C₂²×C₄×C₆₇	Q₁₆	1	4	yes	complemented
102	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₆₇	#41	1	2	yes	—
107	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
108	Q₁₆×C₆₇	—	4	4	no	—
109	C₂×C₈×C₆₇	—	2	1	no	—
110	C₂×C₈×C₆₇	—	2	1	no	—
111	C₂×C₈×C₆₇	—	2	1	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	2	no	—
114	C₂×C₈×C₆₇	—	2	2	no	—
115	C₂×C₈×C₆₇	—	2	2	no	—
116	C₂×C₈×C₆₇	—	4	4	no	—
117	C₂×C₈×C₆₇	—	4	4	no	—
118	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
119	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
120	C₄⋊C₄×C₆₇	—	2	8	no	—
121	C₄⋊C₄×C₆₇	—	2	2	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	—	4	4	no	—
128	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
129	C₂²⋊C₄×C₆₇	—	2	8	no	—
130	C₂²⋊C₄×C₆₇	—	4	4	no	—
131	C₄²×C₆₇	—	2	2	no	—
132	C₄²×C₆₇	—	2	2	no	—
133	C₄²×C₆₇	—	4	2	no	—
134	C₄²×C₆₇	—	4	4	no	—
135	C₄²×C₆₇	—	4	4	no	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	—	2	1	no	—
138	C₂³×C₆₇	—	2	1	no	—
139	C₂³×C₆₇	—	2	1	no	—
140	C₂³×C₆₇	C₄×D₄	1	2	yes	—
141	C₂³×C₆₇	C₂×Q₁₆	1	2	yes	—
142	C₂³×C₆₇	#135	1	2	yes	—
143	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₆₇	—	2	4	no	—
145	Q₈×C₆₇	—	2	4	no	—
146	Q₈×C₆₇	—	4	2	no	—
147	Q₈×C₆₇	—	4	2	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	2	no	—
154	C₂×C₄×C₆₇	—	2	2	no	—
155	C₂×C₄×C₆₇	—	2	2	no	—
156	C₂×C₄×C₆₇	—	2	4	no	—
157	C₂×C₄×C₆₇	—	2	4	no	—
158	C₂×C₄×C₆₇	—	2	2	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	4	4	no	—
162	C₂×C₄×C₆₇	—	4	4	no	—
163	C₂×C₄×C₆₇	—	4	2	no	—
164	C₂×C₄×C₆₇	—	4	2	no	—
165	C₂×C₄×C₆₇	—	4	2	no	—
166	C₂×C₄×C₆₇	—	4	2	no	—
167	C₂×C₄×C₆₇	—	4	2	no	—
168	C₂×C₄×C₆₇	—	4	2	no	—
169	C₂×C₄×C₆₇	—	4	2	no	—
170	C₂×C₄×C₆₇	—	4	2	no	—
171	C₂×C₄×C₆₇	—	4	2	no	—
172	C₂×C₄×C₆₇	—	4	2	no	—
173	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
174	C₂×C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
175	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
177	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
179	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
180	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
181	C₈×C₆₇	—	2	2	no	—
182	C₈×C₆₇	—	2	2	no	—
183	C₈×C₆₇	—	4	2	no	—
184	C₂²×C₆₇	—	2	2	no	—
185	C₂²×C₆₇	—	2	2	no	—
186	C₂²×C₆₇	—	2	2	no	—
187	C₂²×C₆₇	—	2	2	no	—
188	C₂²×C₆₇	—	2	2	no	—
189	C₂²×C₆₇	—	2	2	no	—
190	C₂²×C₆₇	C₄×Q₁₆	1	4	yes	—
191	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
192	C₂²×C₆₇	#534	1	1	yes	characteristic
193	C₂²×C₆₇	#537	1	1	yes	characteristic
194	C₂²×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
195	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
196	C₂²×C₆₇	#575	1	1	yes	characteristic
197	C₂²×C₆₇	#575	1	1	yes	characteristic
198	C₄×C₆₇	—	2	2	no	—
199	C₄×C₆₇	—	2	4	no	—
200	C₄×C₆₇	—	4	4	no	—
201	C₄×C₆₇	—	4	2	no	—
202	C₄×C₆₇	—	4	4	no	—
203	C₄×C₆₇	—	8	2	no	—
204	C₄×C₆₇	—	8	2	no	—
205	C₄×C₆₇	#518	1	1	yes	characteristic
206	C₄×C₆₇	#520	1	1	yes	characteristic
207	C₄×C₆₇	#538	1	1	yes	characteristic
208	C₄×C₆₇	#539	1	1	yes	characteristic
209	#22182	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₆₇	—	2	4	no	—
211	C₂×C₆₇	#2325	1	1	yes	characteristic
212	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
213	C₂×C₆₇	#2943	1	1	yes	characteristic
214	C₂×C₆₇	#3189	1	1	yes	characteristic
215	C₂×C₆₇	#3191	1	1	yes	characteristic
216	C₂×C₆₇	#3249	1	1	yes	characteristic
217	C₂×C₆₇	#3250	1	1	yes	characteristic
218	C₂×#467	C₂×C₆₇	1	1	yes	characteristic
219	C₄×C₂²⋊Q₈	C₂×C₆₇	1	2	yes	—
220	#1928	C₂×C₆₇	1	2	yes	complemented
221	#1921	C₂×C₆₇	1	2	yes	complemented
222	#1892	C₂×C₆₇	1	2	yes	—
223	#1878	C₂×C₆₇	1	2	yes	complemented
224	#1863	C₂×C₆₇	1	2	yes	complemented
225	#1808	C₂×C₆₇	1	1	yes	characteristic
226	C₂×#336	C₂×C₆₇	1	1	yes	characteristic
227	C₆₇	#22182	1	1	yes	characteristic, direct product factor, Sylow
228	C₂²×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
229	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
230	C₂³×C₈	C₂²×C₆₇	1	1	yes	characteristic
231	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
233	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
234	#467	C₄×C₆₇	1	8	yes	complemented
235	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
236	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
237	C₂×#107	C₂²×C₆₇	1	1	yes	characteristic
238	C₂×#103	C₂²×C₆₇	1	2	yes	—
239	C₂×#103	C₂²×C₆₇	1	2	yes	—
240	C₂×#103	C₂²×C₆₇	1	2	yes	—
241	#391	C₂²×C₆₇	1	2	yes	—
242	#389	C₂²×C₆₇	1	2	yes	—
243	#387	—	2	2	no	—
244	#385	—	2	2	no	—
245	#380	C₂²×C₆₇	1	2	yes	—
246	C₄×C₄⋊C₄	—	2	2	no	—
247	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
248	C₄×C₂²⋊C₄	—	2	2	no	—
249	#340	C₂²×C₆₇	1	2	yes	—
250	#340	C₂²×C₆₇	1	2	yes	—
251	#340	C₂²×C₆₇	1	2	yes	—
252	#336	C₂²×C₆₇	1	2	yes	—
253	#336	C₂²×C₆₇	1	2	yes	—
254	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
255	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
256	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
257	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
258	C₂×Q₁₆	—	4	1	no	—
259	C₂×Q₁₆	—	4	1	no	—
260	C₂×Q₁₆	C₂×C₄×C₆₇	1	4	yes	complemented
261	C₂²×C₈	—	2	2	no	—
262	C₂²×C₈	—	4	1	no	—
263	C₂²×C₈	—	4	1	no	—
264	C₂²×C₈	—	4	1	no	—
265	C₂²×C₈	—	4	1	no	—
266	C₂²×C₈	C₂×C₄×C₆₇	1	4	yes	—
267	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
268	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
270	C₂²⋊Q₈	—	2	4	no	—
271	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
272	C₄×Q₈	—	4	2	no	—
273	C₄×Q₈	—	4	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
280	C₂×C₂²⋊C₄	—	2	2	no	—
281	C₂×C₂²⋊C₄	—	2	2	no	—
282	C₂×C₄²	—	2	2	no	—
283	C₂×C₄²	—	2	2	no	—
284	C₂×C₄²	—	4	1	no	—
285	C₂×C₄²	—	4	1	no	—
286	C₂×C₄²	D₄×C₆₇	1	2	yes	—
287	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
288	#107	—	4	1	no	—
289	#107	—	4	1	no	—
290	#107	C₂×C₄×C₆₇	1	4	yes	complemented
291	#103	—	4	2	no	—
292	#103	—	4	2	no	—
293	#103	C₂×C₄×C₆₇	1	8	yes	complemented
294	#95	—	2	2	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	#95	—	2	2	no	—
299	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	2	2	no	—
302	C₂×Q₈	—	2	2	no	—
303	C₂×Q₈	—	2	2	no	—
304	C₂×Q₈	—	2	2	no	—
305	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	2	no	—
315	C₂²×C₄	—	4	1	no	—
316	C₂²×C₄	—	4	1	no	—
317	C₂²×C₄	—	4	1	no	—
318	C₂²×C₄	—	4	1	no	—
319	C₂²×C₄	Q₁₆×C₆₇	1	4	yes	complemented
320	C₂²×C₄	C₂²×C₄×C₆₇	1	2	yes	—
321	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
324	C₂²×C₄	#41×C₆₇	1	2	yes	—
325	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
326	Q₁₆	—	4	4	no	—
327	C₂×C₈	—	2	1	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	4	4	no	—
335	C₂×C₈	—	4	4	no	—
336	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₆₇	1	2	yes	—
338	C₄⋊C₄	—	2	8	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	4	no	—
346	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
347	C₂²⋊C₄	—	2	8	no	—
348	C₂²⋊C₄	—	4	4	no	—
349	C₄²	—	2	2	no	—
350	C₄²	—	2	2	no	—
351	C₄²	—	4	2	no	—
352	C₄²	—	4	4	no	—
353	C₄²	—	4	4	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	1	no	—
357	C₂³	—	2	1	no	—
358	C₂³	C₄×D₄×C₆₇	1	2	yes	—
359	C₂³	C₂×Q₁₆×C₆₇	1	2	yes	—
360	C₂³	#135×C₆₇	1	2	yes	—
361	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
362	Q₈	—	2	4	no	—
363	Q₈	—	2	4	no	—
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	4	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	4	no	—
380	C₂×C₄	—	4	4	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	C₄×D₄×C₆₇	1	2	yes	—
392	C₂×C₄	C₂×Q₁₆×C₆₇	1	4	yes	—
393	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
394	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
395	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
396	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
397	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
398	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
399	C₈	—	2	2	no	—
400	C₈	—	2	2	no	—
401	C₈	—	4	2	no	—
402	C₂²	—	2	2	no	—
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	C₄×Q₁₆×C₆₇	1	4	yes	—
409	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
410	C₂²	#534×C₆₇	1	1	yes	characteristic
411	C₂²	#537×C₆₇	1	1	yes	characteristic
412	C₂²	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
413	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
414	C₂²	#575×C₆₇	1	1	yes	characteristic
415	C₂²	#575×C₆₇	1	1	yes	characteristic
416	C₄	—	2	2	no	—
417	C₄	—	2	4	no	—
418	C₄	—	4	4	no	—
419	C₄	—	4	2	no	—
420	C₄	—	4	4	no	—
421	C₄	—	8	2	no	—
422	C₄	—	8	2	no	—
423	C₄	#518×C₆₇	1	1	yes	characteristic
424	C₄	#520×C₆₇	1	1	yes	characteristic
425	C₄	#538×C₆₇	1	1	yes	characteristic
426	C₄	#539×C₆₇	1	1	yes	characteristic
427	C₂	—	2	4	no	—
428	C₂	C₆₇×#2325	1	1	yes	characteristic
429	C₂	C₂×C₄×Q₁₆×C₆₇	1	1	yes	characteristic
430	C₂	C₆₇×#2943	1	1	yes	characteristic
431	C₂	C₆₇×#3189	1	1	yes	characteristic
432	C₂	C₆₇×#3191	1	1	yes	characteristic
433	C₂	C₆₇×#3249	1	1	yes	characteristic
434	C₂	C₆₇×#3250	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1928	C₂	semidirect
4. C₆₇×#1921	C₂	semidirect
6. C₆₇×#1878	C₂	semidirect
7. C₆₇×#1863	C₂	semidirect
16. #467×C₆₇	C₄	semidirect
42. C₂×Q₁₆×C₆₇	C₂×C₄	semidirect
72. #107×C₆₇	C₂×C₄	semidirect
75. #103×C₆₇	C₂×C₄	semidirect
101. C₂²×C₄×C₆₇	Q₁₆	semidirect
209. #22182	C₆₇	direct (N × H)
220. #1928	C₂×C₆₇	semidirect
221. #1921	C₂×C₆₇	semidirect
223. #1878	C₂×C₆₇	semidirect
224. #1863	C₂×C₆₇	semidirect
227. C₆₇	#22182	direct (N × H)
234. #467	C₄×C₆₇	semidirect
260. C₂×Q₁₆	C₂×C₄×C₆₇	semidirect
290. #107	C₂×C₄×C₆₇	semidirect
293. #103	C₂×C₄×C₆₇	semidirect
319. C₂²×C₄	Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#467×C₆₇	2
C₄×C₂²⋊Q₈×C₆₇	2
C₆₇×#1928	2
C₆₇×#1921	2
C₆₇×#1892	2
C₆₇×#1878	2
C₆₇×#1863	2
C₆₇×#1808	2
C₂×#336×C₆₇	2
#22182	67

Subgroups — C67×#22182 #30701433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22182 #30701433