Subgroups — C₆₇×#32923 #30712174

Order 17152.

1334 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×2₋¹⁺⁴×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#3039	C₂	1	2	yes	maximal, complemented
3	C₂×#456×C₆₇	C₂	1	2	yes	maximal
4	C₆₇×#2546	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1886	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1882	C₂	1	4	yes	maximal, complemented
7	C₆₇×#1806	C₂	1	2	yes	maximal, complemented
8	C₆₇×#1801	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×2₋¹⁺⁴×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×#137×C₆₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	#520×C₆₇	—	2	2	no	—
14	#520×C₆₇	—	2	2	no	—
15	#520×C₆₇	C₂²	1	2	yes	—
16	#519×C₆₇	—	2	2	no	—
17	#519×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×#41×C₆₇	—	2	2	no	—
19	C₄×#41×C₆₇	—	2	2	no	—
20	C₄×#41×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄×Q₈×C₆₇	—	2	2	no	—
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	#462×C₆₇	—	2	2	no	—
24	#462×C₆₇	—	2	2	no	—
25	#461×C₆₇	—	2	2	no	—
26	#461×C₆₇	—	2	2	no	—
27	#456×C₆₇	—	2	8	no	—
28	#423×C₆₇	C₂²	1	2	yes	—
29	C₂×#104×C₆₇	C₂²	1	4	yes	—
30	#419×C₆₇	C₂²	1	4	yes	—
31	C₂×#101×C₆₇	C₂²	1	2	yes	—
32	#398×C₆₇	—	2	2	no	—
33	#391×C₆₇	C₂²	1	2	yes	—
34	#389×C₆₇	—	2	2	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#343×C₆₇	C₂²	1	2	yes	—
37	#339×C₆₇	C₂²	1	4	yes	—
38	2₋¹⁺⁴×C₆₇	—	2	4	no	—
39	2₋¹⁺⁴×C₆₇	—	2	2	no	—
40	2₋¹⁺⁴×C₆₇	D₄	1	4	yes	—
41	C₂×#41×C₆₇	—	2	2	no	—
42	C₂×#41×C₆₇	—	2	2	no	—
43	C₂×#41×C₆₇	C₂³	1	2	yes	—
44	C₂²×Q₈×C₆₇	—	2	2	no	—
45	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
46	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
47	#137×C₆₇	—	2	8	no	—
48	#137×C₆₇	—	2	8	no	—
49	C₂×Q₁₆×C₆₇	—	2	4	no	—
50	C₂×SD₁₆×C₆₇	—	2	4	no	—
51	C₂×M₁₆×C₆₇	C₂³	1	4	yes	—
52	C₄⋊Q₈×C₆₇	—	2	4	no	—
53	C₄⋊Q₈×C₆₇	—	2	4	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄×Q₈×C₆₇	—	2	2	no	—
57	C₄×Q₈×C₆₇	—	2	4	no	—
58	C₄×Q₈×C₆₇	—	2	2	no	—
59	C₄×Q₈×C₆₇	—	2	2	no	—
60	C₄×Q₈×C₆₇	—	2	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	—	2	2	no	—
64	C₄×Q₈×C₆₇	D₄	1	2	yes	—
65	C₄×Q₈×C₆₇	D₄	1	2	yes	—
66	C₄×D₄×C₆₇	—	2	2	no	—
67	C₄×D₄×C₆₇	—	2	4	no	—
68	C₄×D₄×C₆₇	—	2	2	no	—
69	C₄×D₄×C₆₇	—	2	2	no	—
70	C₄×D₄×C₆₇	—	2	2	no	—
71	C₄×D₄×C₆₇	—	2	4	no	—
72	C₄×D₄×C₆₇	—	2	2	no	—
73	C₄×D₄×C₆₇	D₄	1	2	yes	—
74	C₄×D₄×C₆₇	D₄	1	2	yes	—
75	#117×C₆₇	—	2	4	no	—
76	#117×C₆₇	—	2	4	no	—
77	#117×C₆₇	—	2	2	no	—
78	#117×C₆₇	—	2	2	no	—
79	#117×C₆₇	—	2	2	no	—
80	#117×C₆₇	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
82	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
83	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
84	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
85	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
87	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
88	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₆₇	—	2	4	no	—
90	C₂×C₄²×C₆₇	—	2	2	no	—
91	C₂×C₄²×C₆₇	D₄	1	2	yes	—
92	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
93	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
94	C₄⋊C₈×C₆₇	—	2	2	no	—
95	C₄⋊C₈×C₆₇	—	2	2	no	—
96	#104×C₆₇	—	2	8	no	—
97	#103×C₆₇	—	2	4	no	—
98	#103×C₆₇	—	2	4	no	—
99	#101×C₆₇	—	2	4	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	4	no	—
102	#41×C₆₇	—	2	8	no	—
103	#41×C₆₇	—	2	8	no	—
104	#41×C₆₇	—	2	8	no	—
105	#41×C₆₇	—	2	8	no	—
106	#41×C₆₇	—	2	2	no	—
107	#41×C₆₇	—	2	2	no	—
108	#41×C₆₇	C₂×D₄	1	4	yes	—
109	#41×C₆₇	#41	1	4	yes	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	8	no	—
112	C₂×Q₈×C₆₇	—	2	8	no	—
113	C₂×Q₈×C₆₇	—	2	2	no	—
114	C₂×Q₈×C₆₇	—	2	2	no	—
115	C₂×Q₈×C₆₇	—	2	4	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	—	2	2	no	—
118	C₂×Q₈×C₆₇	—	2	4	no	—
119	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
120	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
121	C₂×Q₈×C₆₇	#41	1	2	yes	—
122	C₂×D₄×C₆₇	—	2	2	no	—
123	C₂×D₄×C₆₇	—	2	2	no	—
124	C₂×D₄×C₆₇	C₂×D₄	1	2	yes	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	4	no	—
129	C₂²×C₄×C₆₇	—	2	2	no	—
130	C₂²×C₄×C₆₇	—	2	2	no	—
131	C₂²×C₄×C₆₇	—	2	2	no	—
132	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
137	Q₁₆×C₆₇	—	4	8	no	—
138	SD₁₆×C₆₇	—	4	8	no	—
139	M₁₆×C₆₇	—	2	4	no	—
140	M₁₆×C₆₇	—	2	4	no	—
141	C₂×C₈×C₆₇	—	2	4	no	—
142	C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₄⋊C₄×C₆₇	—	2	4	no	—
144	C₄⋊C₄×C₆₇	—	2	2	no	—
145	C₄⋊C₄×C₆₇	—	2	2	no	—
146	C₄⋊C₄×C₆₇	—	2	2	no	—
147	C₄⋊C₄×C₆₇	—	2	4	no	—
148	C₄⋊C₄×C₆₇	—	2	2	no	—
149	C₄⋊C₄×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	4	1	no	—
151	C₄⋊C₄×C₆₇	—	4	2	no	—
152	C₄⋊C₄×C₆₇	—	4	1	no	—
153	C₄⋊C₄×C₆₇	—	4	8	no	—
154	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
155	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
156	C₂²⋊C₄×C₆₇	—	2	4	no	—
157	C₂²⋊C₄×C₆₇	—	2	2	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₄²×C₆₇	—	2	4	no	—
161	C₄²×C₆₇	—	2	4	no	—
162	C₄²×C₆₇	—	2	2	no	—
163	C₄²×C₆₇	—	2	2	no	—
164	C₄²×C₆₇	—	2	2	no	—
165	C₄²×C₆₇	—	2	2	no	—
166	C₄²×C₆₇	—	2	2	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	C₂×D₄	1	2	yes	—
170	C₄²×C₆₇	C₂×D₄	1	2	yes	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₆₇	—	2	8	no	—
174	Q₈×C₆₇	—	2	4	no	—
175	Q₈×C₆₇	—	2	2	no	—
176	Q₈×C₆₇	—	2	4	no	—
177	Q₈×C₆₇	—	4	4	no	—
178	Q₈×C₆₇	—	4	4	no	—
179	Q₈×C₆₇	#121	1	4	yes	—
180	D₄×C₆₇	—	2	8	no	—
181	D₄×C₆₇	—	2	4	no	—
182	D₄×C₆₇	—	2	4	no	—
183	D₄×C₆₇	—	2	2	no	—
184	D₄×C₆₇	#121	1	4	yes	—
185	C₂×C₄×C₆₇	—	2	16	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	2	no	—
192	C₂×C₄×C₆₇	—	2	4	no	—
193	C₂×C₄×C₆₇	—	2	2	no	—
194	C₂×C₄×C₆₇	—	2	2	no	—
195	C₂×C₄×C₆₇	—	2	2	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	2	no	—
199	C₂×C₄×C₆₇	—	4	4	no	—
200	C₂×C₄×C₆₇	—	4	2	no	—
201	C₂×C₄×C₆₇	—	4	2	no	—
202	C₂×C₄×C₆₇	—	4	2	no	—
203	C₂×C₄×C₆₇	—	4	2	no	—
204	C₂×C₄×C₆₇	—	8	2	no	—
205	C₂×C₄×C₆₇	—	8	4	no	—
206	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
207	C₂×C₄×C₆₇	#120	1	2	yes	—
208	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
209	C₂×C₄×C₆₇	#121	1	4	yes	—
210	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
213	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
214	C₈×C₆₇	—	4	4	no	—
215	C₂²×C₆₇	—	2	1	no	—
216	C₂²×C₆₇	—	2	1	no	—
217	C₂²×C₆₇	—	2	4	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
221	C₂²×C₆₇	C₂×#121	1	1	yes	characteristic
222	C₂²×C₆₇	#525	1	1	yes	characteristic
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	2	1	no	—
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	4	4	no	—
228	C₄×C₆₇	—	4	2	no	—
229	C₄×C₆₇	—	4	2	no	—
230	C₄×C₆₇	—	4	2	no	—
231	C₄×C₆₇	—	8	2	no	—
232	C₄×C₆₇	#456	1	2	yes	complemented
233	C₄×C₆₇	C₂×#121	1	2	yes	—
234	C₄×C₆₇	#525	1	2	yes	—
235	#32923	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
236	C₂×C₆₇	—	2	1	no	—
237	C₂×C₆₇	—	2	1	no	—
238	C₂×C₆₇	—	4	4	no	—
239	C₂×C₆₇	#2847	1	1	yes	characteristic
240	C₂×C₆₇	C₂×#456	1	1	yes	characteristic
241	C₂×C₆₇	#3022	1	1	yes	characteristic
242	C₄×2₋¹⁺⁴	C₂×C₆₇	1	1	yes	characteristic
243	#3039	C₂×C₆₇	1	2	yes	complemented
244	C₂×#456	C₂×C₆₇	1	2	yes	—
245	#2546	C₂×C₆₇	1	1	yes	characteristic, complemented
246	#1886	C₂×C₆₇	1	2	yes	complemented
247	#1882	C₂×C₆₇	1	4	yes	complemented
248	#1806	C₂×C₆₇	1	2	yes	complemented
249	#1801	C₂×C₆₇	1	1	yes	characteristic, complemented
250	C₆₇	#32923	1	1	yes	characteristic, direct product factor, Sylow
251	C₂×2₋¹⁺⁴	C₂²×C₆₇	1	1	yes	characteristic
252	C₂×#137	C₂²×C₆₇	1	4	yes	—
253	C₂×C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
254	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
255	#520	—	2	2	no	—
256	#520	—	2	2	no	—
257	#520	C₂²×C₆₇	1	2	yes	—
258	#519	—	2	2	no	—
259	#519	C₂²×C₆₇	1	1	yes	characteristic
260	C₄×#41	—	2	2	no	—
261	C₄×#41	—	2	2	no	—
262	C₄×#41	C₂²×C₆₇	1	2	yes	—
263	C₂×C₄×Q₈	—	2	2	no	—
264	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
265	#462	—	2	2	no	—
266	#462	—	2	2	no	—
267	#461	—	2	2	no	—
268	#461	—	2	2	no	—
269	#456	—	2	8	no	—
270	#423	C₂²×C₆₇	1	2	yes	—
271	C₂×#104	C₂²×C₆₇	1	4	yes	—
272	#419	C₂²×C₆₇	1	4	yes	—
273	C₂×#101	C₂²×C₆₇	1	2	yes	—
274	#398	—	2	2	no	—
275	#391	C₂²×C₆₇	1	2	yes	—
276	#389	—	2	2	no	—
277	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
278	#343	C₂²×C₆₇	1	2	yes	—
279	#339	C₂²×C₆₇	1	4	yes	—
280	2₋¹⁺⁴	—	2	4	no	—
281	2₋¹⁺⁴	—	2	2	no	—
282	2₋¹⁺⁴	D₄×C₆₇	1	4	yes	—
283	C₂×#41	—	2	2	no	—
284	C₂×#41	—	2	2	no	—
285	C₂×#41	C₂³×C₆₇	1	2	yes	—
286	C₂²×Q₈	—	2	2	no	—
287	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
288	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
289	#137	—	2	8	no	—
290	#137	—	2	8	no	—
291	C₂×Q₁₆	—	2	4	no	—
292	C₂×SD₁₆	—	2	4	no	—
293	C₂×M₁₆	C₂³×C₆₇	1	4	yes	—
294	C₄⋊Q₈	—	2	4	no	—
295	C₄⋊Q₈	—	2	4	no	—
296	C₄⋊Q₈	—	2	2	no	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄×Q₈	—	2	2	no	—
299	C₄×Q₈	—	2	4	no	—
300	C₄×Q₈	—	2	2	no	—
301	C₄×Q₈	—	2	2	no	—
302	C₄×Q₈	—	2	2	no	—
303	C₄×Q₈	—	2	2	no	—
304	C₄×Q₈	—	2	2	no	—
305	C₄×Q₈	—	2	2	no	—
306	C₄×Q₈	D₄×C₆₇	1	2	yes	—
307	C₄×Q₈	D₄×C₆₇	1	2	yes	—
308	C₄×D₄	—	2	2	no	—
309	C₄×D₄	—	2	4	no	—
310	C₄×D₄	—	2	2	no	—
311	C₄×D₄	—	2	2	no	—
312	C₄×D₄	—	2	2	no	—
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	2	2	no	—
315	C₄×D₄	D₄×C₆₇	1	2	yes	—
316	C₄×D₄	D₄×C₆₇	1	2	yes	—
317	#117	—	2	4	no	—
318	#117	—	2	4	no	—
319	#117	—	2	2	no	—
320	#117	—	2	2	no	—
321	#117	—	2	2	no	—
322	#117	C₂³×C₆₇	1	2	yes	—
323	C₂×C₄⋊C₄	—	2	4	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	2	no	—
326	C₂×C₄⋊C₄	—	2	1	no	—
327	C₂×C₄⋊C₄	—	2	4	no	—
328	C₂×C₄⋊C₄	—	2	1	no	—
329	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
330	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
331	C₂×C₄²	—	2	4	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	D₄×C₆₇	1	2	yes	—
334	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
335	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
336	C₄⋊C₈	—	2	2	no	—
337	C₄⋊C₈	—	2	2	no	—
338	#104	—	2	8	no	—
339	#103	—	2	4	no	—
340	#103	—	2	4	no	—
341	#101	—	2	4	no	—
342	#95	—	2	1	no	—
343	#95	—	2	4	no	—
344	#41	—	2	8	no	—
345	#41	—	2	8	no	—
346	#41	—	2	8	no	—
347	#41	—	2	8	no	—
348	#41	—	2	2	no	—
349	#41	—	2	2	no	—
350	#41	C₂×D₄×C₆₇	1	4	yes	—
351	#41	#41×C₆₇	1	4	yes	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	8	no	—
354	C₂×Q₈	—	2	8	no	—
355	C₂×Q₈	—	2	2	no	—
356	C₂×Q₈	—	2	2	no	—
357	C₂×Q₈	—	2	4	no	—
358	C₂×Q₈	—	2	2	no	—
359	C₂×Q₈	—	2	2	no	—
360	C₂×Q₈	—	2	4	no	—
361	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
362	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
363	C₂×Q₈	#41×C₆₇	1	2	yes	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	—	2	2	no	—
366	C₂×D₄	C₂×D₄×C₆₇	1	2	yes	—
367	C₂²×C₄	—	2	4	no	—
368	C₂²×C₄	—	2	2	no	—
369	C₂²×C₄	—	2	2	no	—
370	C₂²×C₄	—	2	4	no	—
371	C₂²×C₄	—	2	2	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	2	no	—
374	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
378	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
379	Q₁₆	—	4	8	no	—
380	SD₁₆	—	4	8	no	—
381	M₁₆	—	2	4	no	—
382	M₁₆	—	2	4	no	—
383	C₂×C₈	—	2	4	no	—
384	C₄⋊C₄	—	2	4	no	—
385	C₄⋊C₄	—	2	2	no	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	2	no	—
389	C₄⋊C₄	—	2	4	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	—	4	1	no	—
393	C₄⋊C₄	—	4	2	no	—
394	C₄⋊C₄	—	4	1	no	—
395	C₄⋊C₄	—	4	8	no	—
396	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
397	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
398	C₂²⋊C₄	—	2	4	no	—
399	C₂²⋊C₄	—	2	2	no	—
400	C₂²⋊C₄	—	2	2	no	—
401	C₂²⋊C₄	—	2	2	no	—
402	C₄²	—	2	4	no	—
403	C₄²	—	2	4	no	—
404	C₄²	—	2	2	no	—
405	C₄²	—	2	2	no	—
406	C₄²	—	2	2	no	—
407	C₄²	—	2	2	no	—
408	C₄²	—	2	2	no	—
409	C₄²	—	2	2	no	—
410	C₄²	—	2	2	no	—
411	C₄²	C₂×D₄×C₆₇	1	2	yes	—
412	C₄²	C₂×D₄×C₆₇	1	2	yes	—
413	C₂³	—	2	2	no	—
414	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
415	Q₈	—	2	8	no	—
416	Q₈	—	2	4	no	—
417	Q₈	—	2	2	no	—
418	Q₈	—	2	4	no	—
419	Q₈	—	4	4	no	—
420	Q₈	—	4	4	no	—
421	Q₈	#121×C₆₇	1	4	yes	—
422	D₄	—	2	8	no	—
423	D₄	—	2	4	no	—
424	D₄	—	2	4	no	—
425	D₄	—	2	2	no	—
426	D₄	#121×C₆₇	1	4	yes	—
427	C₂×C₄	—	2	16	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	4	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	2	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	4	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	2	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	8	2	no	—
447	C₂×C₄	—	8	4	no	—
448	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
449	C₂×C₄	#120×C₆₇	1	2	yes	—
450	C₂×C₄	#120×C₆₇	1	1	yes	characteristic
451	C₂×C₄	#121×C₆₇	1	4	yes	—
452	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
454	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
455	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
456	C₈	—	4	4	no	—
457	C₂²	—	2	1	no	—
458	C₂²	—	2	1	no	—
459	C₂²	—	2	4	no	—
460	C₂²	—	4	2	no	—
461	C₂²	—	4	2	no	—
462	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
463	C₂²	C₂×#121×C₆₇	1	1	yes	characteristic
464	C₂²	#525×C₆₇	1	1	yes	characteristic
465	C₄	—	2	1	no	—
466	C₄	—	2	2	no	—
467	C₄	—	2	4	no	—
468	C₄	—	2	4	no	—
469	C₄	—	4	4	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	8	2	no	—
474	C₄	#456×C₆₇	1	2	yes	complemented
475	C₄	C₂×#121×C₆₇	1	2	yes	—
476	C₄	#525×C₆₇	1	2	yes	—
477	C₂	—	2	1	no	—
478	C₂	—	2	1	no	—
479	C₂	—	4	4	no	—
480	C₂	C₆₇×#2847	1	1	yes	characteristic
481	C₂	C₂×#456×C₆₇	1	1	yes	characteristic
482	C₂	C₆₇×#3022	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂×2₋¹⁺⁴×C₆₇	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₆₇×#3039	C₂	semidirect
4. C₆₇×#2546	C₂	semidirect
5. C₆₇×#1886	C₂	semidirect
6. C₆₇×#1882	C₂	semidirect
7. C₆₇×#1806	C₂	semidirect
8. C₆₇×#1801	C₂	semidirect
232. C₄×C₆₇	#456	semidirect
235. #32923	C₆₇	direct (N × H)
243. #3039	C₂×C₆₇	semidirect
245. #2546	C₂×C₆₇	semidirect
246. #1886	C₂×C₆₇	semidirect
247. #1882	C₂×C₆₇	semidirect
248. #1806	C₂×C₆₇	semidirect
249. #1801	C₂×C₆₇	semidirect
250. C₆₇	#32923	direct (N × H)
474. C₄	#456×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×2₋¹⁺⁴×C₆₇	2
C₆₇×#3039	2
C₂×#456×C₆₇	2
C₆₇×#2546	2
C₆₇×#1886	2
C₆₇×#1882	2
C₆₇×#1806	2
C₆₇×#1801	2
#32923	67

Subgroups — C67×#32923 #30712174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32923 #30712174