Subgroups — C₆₇×#26020 #30705271

Order 17152.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2551	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2354	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#2067	C₂	1	2	yes	maximal
4	C₆₇×#2023	C₂	1	2	yes	maximal
5	C₆₇×#1988	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1918	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1834	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₆₇	—	2	1	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#117×C₆₇	—	2	1	no	—
16	C₂×#107×C₆₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
19	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#102×C₆₇	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	#395×C₆₇	—	2	2	no	—
24	#390×C₆₇	C₂²	1	2	yes	—
25	#389×C₆₇	—	2	1	no	—
26	#389×C₆₇	C₂²	1	2	yes	—
27	#388×C₆₇	—	2	2	no	—
28	#384×C₆₇	C₂²	1	1	yes	characteristic
29	#384×C₆₇	C₂²	1	1	yes	characteristic
30	#380×C₆₇	—	2	1	no	—
31	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
33	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
34	#340×C₆₇	C₂²	1	1	yes	characteristic
35	#340×C₆₇	C₂²	1	2	yes	—
36	#340×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	1	yes	characteristic
39	#340×C₆₇	C₂²	1	2	yes	—
40	#336×C₆₇	C₂²	1	2	yes	—
41	C₂²×Q₈×C₆₇	—	2	2	no	—
42	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₆₇	—	2	1	no	—
44	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₆₇	—	2	4	no	—
48	C₄⋊Q₈×C₆₇	—	4	2	no	—
49	C₄⋊Q₈×C₆₇	—	4	4	no	—
50	C₄×D₄×C₆₇	—	2	2	no	—
51	C₄×D₄×C₆₇	Q₈	1	4	yes	complemented
52	#117×C₆₇	—	4	4	no	—
53	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
54	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
55	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
56	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
57	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
58	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
59	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
69	C₂×C₄²×C₆₇	—	2	2	no	—
70	C₂×C₄²×C₆₇	—	2	1	no	—
71	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#107×C₆₇	—	2	4	no	—
74	C₄⋊C₈×C₆₇	—	2	2	no	—
75	C₄⋊C₈×C₆₇	—	4	2	no	—
76	#102×C₆₇	—	2	2	no	—
77	#102×C₆₇	—	2	2	no	—
78	#102×C₆₇	—	4	2	no	—
79	C₄×C₈×C₆₇	—	2	2	no	—
80	#95×C₆₇	—	2	2	no	—
81	#95×C₆₇	—	2	1	no	—
82	#95×C₆₇	—	2	2	no	—
83	#95×C₆₇	D₄	1	2	yes	—
84	C₂⁴×C₆₇	—	2	1	no	—
85	C₂×Q₈×C₆₇	—	4	4	no	—
86	C₂×Q₈×C₆₇	—	4	4	no	—
87	C₂×Q₈×C₆₇	—	8	2	no	—
88	C₂×D₄×C₆₇	—	2	1	no	—
89	C₂×D₄×C₆₇	—	2	1	no	—
90	C₂×D₄×C₆₇	—	2	1	no	—
91	C₂×D₄×C₆₇	—	2	1	no	—
92	C₂×D₄×C₆₇	—	2	1	no	—
93	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
94	C₂²×C₄×C₆₇	—	2	1	no	—
95	C₂²×C₄×C₆₇	—	2	1	no	—
96	C₂²×C₄×C₆₇	—	2	1	no	—
97	C₂²×C₄×C₆₇	—	2	4	no	—
98	C₂²×C₄×C₆₇	—	2	2	no	—
99	C₂²×C₄×C₆₇	—	2	2	no	—
100	C₂²×C₄×C₆₇	—	4	2	no	—
101	C₂²×C₄×C₆₇	—	4	2	no	—
102	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₆₇	—	2	1	no	—
108	C₂×C₈×C₆₇	—	2	1	no	—
109	C₂×C₈×C₆₇	—	2	2	no	—
110	C₂×C₈×C₆₇	—	2	2	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	1	no	—
115	C₂×C₈×C₆₇	—	2	1	no	—
116	C₄⋊C₄×C₆₇	—	2	2	no	—
117	C₄⋊C₄×C₆₇	—	2	1	no	—
118	C₄⋊C₄×C₆₇	—	2	1	no	—
119	C₄⋊C₄×C₆₇	—	2	1	no	—
120	C₄⋊C₄×C₆₇	—	2	2	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	—	4	4	no	—
125	C₄⋊C₄×C₆₇	—	4	2	no	—
126	C₄⋊C₄×C₆₇	—	4	4	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	8	1	no	—
129	C₄⋊C₄×C₆₇	—	8	1	no	—
130	C₄⋊C₄×C₆₇	D₈	1	4	yes	—
131	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
132	C₄⋊C₄×C₆₇	#41	1	2	yes	—
133	C₄⋊C₄×C₆₇	#41	1	2	yes	—
134	C₂²⋊C₄×C₆₇	—	2	4	no	—
135	C₂²⋊C₄×C₆₇	—	4	4	no	—
136	C₂²⋊C₄×C₆₇	—	8	2	no	—
137	C₄²×C₆₇	—	2	2	no	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	—	4	4	no	—
140	C₄²×C₆₇	—	4	2	no	—
141	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
142	C₄²×C₆₇	#41	1	2	yes	—
143	C₂³×C₆₇	—	2	2	no	—
144	C₂³×C₆₇	—	4	1	no	—
145	C₂³×C₆₇	—	4	1	no	—
146	C₂³×C₆₇	—	4	1	no	—
147	C₂³×C₆₇	—	4	1	no	—
148	C₂³×C₆₇	—	4	1	no	—
149	C₂³×C₆₇	—	4	1	no	—
150	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	Q₈×C₆₇	—	8	8	no	—
152	D₄×C₆₇	—	2	2	no	—
153	D₄×C₆₇	—	2	2	no	—
154	D₄×C₆₇	—	4	1	no	—
155	D₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	2	2	no	—
157	C₂×C₄×C₆₇	—	2	1	no	—
158	C₂×C₄×C₆₇	—	2	1	no	—
159	C₂×C₄×C₆₇	—	2	2	no	—
160	C₂×C₄×C₆₇	—	2	2	no	—
161	C₂×C₄×C₆₇	—	2	2	no	—
162	C₂×C₄×C₆₇	—	2	2	no	—
163	C₂×C₄×C₆₇	—	2	2	no	—
164	C₂×C₄×C₆₇	—	2	2	no	—
165	C₂×C₄×C₆₇	—	2	2	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	—	4	1	no	—
168	C₂×C₄×C₆₇	—	4	1	no	—
169	C₂×C₄×C₆₇	—	4	1	no	—
170	C₂×C₄×C₆₇	—	4	1	no	—
171	C₂×C₄×C₆₇	—	4	1	no	—
172	C₂×C₄×C₆₇	—	4	8	no	—
173	C₂×C₄×C₆₇	—	4	2	no	—
174	C₂×C₄×C₆₇	—	4	2	no	—
175	C₂×C₄×C₆₇	—	4	2	no	—
176	C₂×C₄×C₆₇	—	4	2	no	—
177	C₂×C₄×C₆₇	—	4	2	no	—
178	C₂×C₄×C₆₇	—	4	2	no	—
179	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
180	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
181	C₂×C₄×C₆₇	C₂×D₈	1	4	yes	—
182	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
183	C₂×C₄×C₆₇	#137	1	2	yes	—
184	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
187	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
188	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
189	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
190	C₈×C₆₇	—	4	1	no	—
191	C₈×C₆₇	—	4	2	no	—
192	C₈×C₆₇	—	4	1	no	—
193	C₂²×C₆₇	—	4	2	no	—
194	C₂²×C₆₇	—	4	2	no	—
195	C₂²×C₆₇	—	4	2	no	—
196	C₂²×C₆₇	—	8	1	no	—
197	C₂²×C₆₇	—	8	1	no	—
198	C₂²×C₆₇	—	8	1	no	—
199	C₂²×C₆₇	—	8	1	no	—
200	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
201	C₂²×C₆₇	#546	1	1	yes	characteristic
202	C₂²×C₆₇	#548	1	1	yes	characteristic
203	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
204	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
205	C₂²×C₆₇	#577	1	1	yes	characteristic
206	C₂²×C₆₇	#577	1	1	yes	characteristic
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	4	2	no	—
210	C₄×C₆₇	—	4	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	8	2	no	—
214	C₄×C₆₇	—	8	2	no	—
215	C₄×C₆₇	#474	1	4	yes	—
216	C₄×C₆₇	#533	1	1	yes	characteristic
217	C₄×C₆₇	#544	1	1	yes	characteristic
218	C₄×C₆₇	#555	1	1	yes	characteristic
219	C₄×C₆₇	#556	1	1	yes	characteristic
220	#26020	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
221	C₂×C₆₇	—	8	2	no	—
222	C₂×C₆₇	#2439	1	1	yes	characteristic
223	C₂×C₆₇	C₂×#474	1	1	yes	characteristic
224	C₂×C₆₇	#3084	1	1	yes	characteristic
225	C₂×C₆₇	#3294	1	1	yes	characteristic
226	C₂×C₆₇	#3329	1	1	yes	characteristic
227	C₂×C₆₇	#3358	1	1	yes	characteristic
228	C₂×C₆₇	#3369	1	1	yes	characteristic
229	#2551	C₂×C₆₇	1	1	yes	characteristic, complemented
230	#2354	C₂×C₆₇	1	1	yes	characteristic
231	#2067	C₂×C₆₇	1	2	yes	—
232	#2023	C₂×C₆₇	1	2	yes	—
233	#1988	C₂×C₆₇	1	2	yes	complemented
234	#1947	C₂×C₆₇	1	1	yes	characteristic, complemented
235	#1939	C₂×C₆₇	1	1	yes	characteristic, complemented
236	#1918	C₂×C₆₇	1	2	yes	complemented
237	#1834	C₂×C₆₇	1	1	yes	characteristic, complemented
238	#1802	C₂×C₆₇	1	1	yes	characteristic
239	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
240	C₆₇	#26020	1	1	yes	characteristic, direct product factor, Sylow
241	C₂×C₄⋊Q₈	—	2	1	no	—
242	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
243	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
244	C₂×#117	—	2	1	no	—
245	C₂×#107	C₂²×C₆₇	1	2	yes	—
246	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
247	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
248	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
249	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
250	C₂×#102	C₂²×C₆₇	1	2	yes	—
251	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
252	#395	—	2	2	no	—
253	#390	C₂²×C₆₇	1	2	yes	—
254	#389	—	2	1	no	—
255	#389	C₂²×C₆₇	1	2	yes	—
256	#388	—	2	2	no	—
257	#384	C₂²×C₆₇	1	1	yes	characteristic
258	#384	C₂²×C₆₇	1	1	yes	characteristic
259	#380	—	2	1	no	—
260	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
261	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
262	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
263	#340	C₂²×C₆₇	1	2	yes	—
264	#340	C₂²×C₆₇	1	2	yes	—
265	#340	C₂²×C₆₇	1	2	yes	—
266	#340	C₂²×C₆₇	1	1	yes	characteristic
267	#340	C₂²×C₆₇	1	2	yes	—
268	#340	C₂²×C₆₇	1	1	yes	characteristic
269	#336	C₂²×C₆₇	1	2	yes	—
270	C₂²×Q₈	—	2	2	no	—
271	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
272	C₂³×C₄	—	2	1	no	—
273	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
274	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
275	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
276	C₄⋊Q₈	—	2	4	no	—
277	C₄⋊Q₈	—	4	2	no	—
278	C₄⋊Q₈	—	4	4	no	—
279	C₄×D₄	—	2	2	no	—
280	C₄×D₄	Q₈×C₆₇	1	4	yes	complemented
281	#117	—	4	4	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	—	2	2	no	—
285	C₂×C₄⋊C₄	—	2	2	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
289	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
291	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
292	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
293	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
295	C₂×C₂²⋊C₄	—	2	1	no	—
296	C₂×C₂²⋊C₄	—	2	2	no	—
297	C₂×C₂²⋊C₄	—	2	2	no	—
298	C₂×C₄²	—	2	2	no	—
299	C₂×C₄²	—	2	1	no	—
300	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
301	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
302	#107	—	2	4	no	—
303	C₄⋊C₈	—	2	2	no	—
304	C₄⋊C₈	—	4	2	no	—
305	#102	—	2	2	no	—
306	#102	—	2	2	no	—
307	#102	—	4	2	no	—
308	C₄×C₈	—	2	2	no	—
309	#95	—	2	2	no	—
310	#95	—	2	1	no	—
311	#95	—	2	2	no	—
312	#95	D₄×C₆₇	1	2	yes	—
313	C₂⁴	—	2	1	no	—
314	C₂×Q₈	—	4	4	no	—
315	C₂×Q₈	—	4	4	no	—
316	C₂×Q₈	—	8	2	no	—
317	C₂×D₄	—	2	1	no	—
318	C₂×D₄	—	2	1	no	—
319	C₂×D₄	—	2	1	no	—
320	C₂×D₄	—	2	1	no	—
321	C₂×D₄	—	2	1	no	—
322	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
323	C₂²×C₄	—	2	1	no	—
324	C₂²×C₄	—	2	1	no	—
325	C₂²×C₄	—	2	1	no	—
326	C₂²×C₄	—	2	4	no	—
327	C₂²×C₄	—	2	2	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	4	2	no	—
330	C₂²×C₄	—	4	2	no	—
331	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
332	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
333	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
334	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
335	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
336	C₂×C₈	—	2	1	no	—
337	C₂×C₈	—	2	1	no	—
338	C₂×C₈	—	2	1	no	—
339	C₂×C₈	—	2	1	no	—
340	C₂×C₈	—	2	2	no	—
341	C₂×C₈	—	2	2	no	—
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	1	no	—
344	C₂×C₈	—	2	1	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	2	1	no	—
347	C₄⋊C₄	—	2	1	no	—
348	C₄⋊C₄	—	2	1	no	—
349	C₄⋊C₄	—	2	2	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	4	4	no	—
354	C₄⋊C₄	—	4	2	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	8	1	no	—
358	C₄⋊C₄	—	8	1	no	—
359	C₄⋊C₄	D₈×C₆₇	1	4	yes	—
360	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
361	C₄⋊C₄	#41×C₆₇	1	2	yes	—
362	C₄⋊C₄	#41×C₆₇	1	2	yes	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	4	4	no	—
365	C₂²⋊C₄	—	8	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	4	4	no	—
369	C₄²	—	4	2	no	—
370	C₄²	C₂×Q₈×C₆₇	1	2	yes	—
371	C₄²	#41×C₆₇	1	2	yes	—
372	C₂³	—	2	2	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	1	no	—
377	C₂³	—	4	1	no	—
378	C₂³	—	4	1	no	—
379	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
380	Q₈	—	8	8	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	2	no	—
383	D₄	—	4	1	no	—
384	D₄	—	4	1	no	—
385	C₂×C₄	—	2	2	no	—
386	C₂×C₄	—	2	1	no	—
387	C₂×C₄	—	2	1	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	2	2	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	4	1	no	—
399	C₂×C₄	—	4	1	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	8	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	4	2	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
409	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
410	C₂×C₄	C₂×D₈×C₆₇	1	4	yes	—
411	C₂×C₄	C₂×D₈×C₆₇	1	2	yes	—
412	C₂×C₄	#137×C₆₇	1	2	yes	—
413	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
415	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
416	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
417	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
418	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
419	C₈	—	4	1	no	—
420	C₈	—	4	2	no	—
421	C₈	—	4	1	no	—
422	C₂²	—	4	2	no	—
423	C₂²	—	4	2	no	—
424	C₂²	—	4	2	no	—
425	C₂²	—	8	1	no	—
426	C₂²	—	8	1	no	—
427	C₂²	—	8	1	no	—
428	C₂²	—	8	1	no	—
429	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
430	C₂²	#546×C₆₇	1	1	yes	characteristic
431	C₂²	#548×C₆₇	1	1	yes	characteristic
432	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
433	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
434	C₂²	#577×C₆₇	1	1	yes	characteristic
435	C₂²	#577×C₆₇	1	1	yes	characteristic
436	C₄	—	2	4	no	—
437	C₄	—	2	2	no	—
438	C₄	—	4	2	no	—
439	C₄	—	4	2	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	2	no	—
442	C₄	—	8	2	no	—
443	C₄	—	8	2	no	—
444	C₄	#474×C₆₇	1	4	yes	—
445	C₄	#533×C₆₇	1	1	yes	characteristic
446	C₄	#544×C₆₇	1	1	yes	characteristic
447	C₄	#555×C₆₇	1	1	yes	characteristic
448	C₄	#556×C₆₇	1	1	yes	characteristic
449	C₂	—	8	2	no	—
450	C₂	C₆₇×#2439	1	1	yes	characteristic
451	C₂	C₂×#474×C₆₇	1	1	yes	characteristic
452	C₂	C₆₇×#3084	1	1	yes	characteristic
453	C₂	C₆₇×#3294	1	1	yes	characteristic
454	C₂	C₆₇×#3329	1	1	yes	characteristic
455	C₂	C₆₇×#3358	1	1	yes	characteristic
456	C₂	C₆₇×#3369	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2551	C₂	semidirect
5. C₆₇×#1988	C₂	semidirect
6. C₆₇×#1947	C₂	semidirect
7. C₆₇×#1939	C₂	semidirect
8. C₆₇×#1918	C₂	semidirect
9. C₆₇×#1834	C₂	semidirect
51. C₄×D₄×C₆₇	Q₈	semidirect
220. #26020	C₆₇	direct (N × H)
229. #2551	C₂×C₆₇	semidirect
233. #1988	C₂×C₆₇	semidirect
234. #1947	C₂×C₆₇	semidirect
235. #1939	C₂×C₆₇	semidirect
236. #1918	C₂×C₆₇	semidirect
237. #1834	C₂×C₆₇	semidirect
240. C₆₇	#26020	direct (N × H)
280. C₄×D₄	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2551	2
C₆₇×#2354	2
C₆₇×#2067	2
C₆₇×#2023	2
C₆₇×#1988	2
C₆₇×#1947	2
C₆₇×#1939	2
C₆₇×#1918	2
C₆₇×#1834	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#26020	67

Subgroups — C67×#26020 #30705271

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#26020 #30705271